The #1 AI tutor AI Learning Service

AI tutor
AI tutor
Join our Discord

Connect With Us on Social Media

AI tutorAI tutorAI tutorAI tutor

AI tutor | The No.1 Homework Finishing Free App

Numbers and Algebra

Fundamental Algebra - Expansion and Factorization of Expressions

Q.01

'The general term of the expansion is\n\\[\\frac{6!}{p!q!r!} \\cdot a^{p} \\cdot(2 b)^{q} \\cdot(3 c)^{r}=\\frac{6!}{p!q!r!} \\cdot 2^{q} \\cdot 3^{r} \\cdot a^{p} b^{q} c^{r}\\]\nwhere \ \\quad p+q+r=6, p \\geqq 0, q \\geqq 0, r \\geqq 0 \\n(a) The coefficient of the term \ a^{3} b^{2} c \ is, when \ p=3, q=2, r=1 \,\n\\\frac{6!}{3!2!1!} \\cdot 2^{2} \\cdot 3^{1}=720\\n(b) The coefficient of the term \ a^{4} c^{2} \ is, when \ p=4, q=0, r=2 \,\n\\\frac{6!}{4!0!2!} \\cdot 2^{0} \\cdot 3^{2}=135\'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.02

'Find the coefficient of the specified term in the following expansion expressions.(1) (2x-y-3z)^6 [xy^3 z^2] (2) (1+x+x^2)^10 [x^4] (3) (x+1/x^2+1)^5 [constant term]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.03

'Find the general term and coefficient of the specific terms for the following expressions:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.04

'(1) \\((x+2-i)(x+2+i)\\)(2) \\((3 x-17)(2 x-9)\\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.05

'Solve the following equations:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.06

'The general term of the expansion \\( (a+b+c)^{n} \\) is\n\\\frac{n!}{p!q!r!} \\alpha^{p} b^{q} c^{r}\\nwhere \ p+q+r=n \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.07

'(2) (Solution 1) α^{3}+β^{3}+γ^{3}=(α+β+γ){α^{2}+β^{2}+γ^{2}-(αβ+βγ+γα)}+3αβγ =2 \\cdot(4-0)+3\\cdot4=20'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.08

'(2) The general term of the expansion is frac10!p!q!r!cdot1pcdotxqcdotleft(x2right)r=frac10!p!q!r!cdotxq+2r\\frac{10!}{p!q!r!} \\cdot 1^{p} \\cdot x^{q} \\cdot\\left(x^{2}\\right)^{r}=\\frac{10!}{p!q!r!} \\cdot x^{q+2 r} where p+q+r=10quadcdotscdots(1),pgeqq0,qgeqq0,rgeqq0p+q+r=10 \\quad\\cdots\\cdots (1), p \\geqq 0, q \\geqq 0, r \\geqq 0.\nThe term with x4x^{4} occurs when q+2r=4q+2 r=4, that is when q=42rq=4-2r.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.09

'Find the general term of the following sequence \ \\left\\{a_{n}\\right\\} \.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.10

'Expand the following polynomials.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.11

'Summation symbol \ \\Sigma \, Properties of \ \\Sigma \\nSummation symbol \ \\Sigma \\n\\n\\sum_{k=1}^{n} a_{k}=a_{1}+a_{2}+a_{3}+\\cdots \\cdots+a_{n}\n\\nThe constants \ p, q \ in this property are independent of \ k \.\n\\[\n\\sum_{k=1}^{n}\\left(p a_{k}+q b_{k}\\right)=p \\sum_{k=1}^{n} a_{k}+q \\sum_{k=1}^{n} b_{k}\n\\]\nThe constants \ c, r \ in the formulas for sums of sequences are independent of \ n \.\n\\[\n\egin{aligned}\n\\sum_{k=1}^{n} c & =n c \\\\ \nIn particular \\\\ \n\\sum_{k=1}^{n} 1=n \\\\ \n\\sum_{k=1}^{n} k & =\\frac{1}{2} n(n+1) \\\\ \n\\sum_{k=1}^{n} k^{2} & =\\frac{1}{6} n(n+1)(2 n+1) \\\\ \n\\sum_{k=1}^{n} k^{3} & =\\left\\{\\frac{1}{2} n(n+1)\\right\\}^{2} \\\\ \n\\sum_{k=1}^{n} r^{k-1} & =\\frac{1-r^{n}}{1-r} \\\\( r \\neq 1) \n\\end{aligned}\\]\n'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.12

'Application problems of binomial theorem'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.13

'(A + B)(A - B)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.14

'(1) 6x² + 3 (2) −x⁸ + 3x³ − 1 (3) 2(x² + 1)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.15

'Translate the given text into multiple languages.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.16

'(A + B)^2 + (A - B)^2'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.17

'Simplify the following continued fraction:\n\\n\\frac{1}{1+\\frac{1}{1+\\frac{1}{x+1}}}\n\'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.18

'Expanding and simplifying the equation obtained in (3) (2) gives: x^2 - mx + y^2 - (m^2 + 2)y = 0. Substituting y = x^2 gives x^2 - mx + x^4 - (m^2 + 2)x^2 = 0, which simplifies to x(x + m)(x^2 - mx - 1) = 0. Therefore, x = 0, -m, α, β. Thus, the necessary and sufficient condition for the parabola y = x^2 and the circle obtained in (2) A, B, O to have no other common points is for x = -m to be a root of the equation x(x^2 - mx - 1) = 0.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.19

'Factorize the following quadratic equations in the range of complex numbers:\n1. x^{2}+4 x+5\n2. 6 x^{2}-61 x+153'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.20

'(2) 1 + 3x + 5x^{2} + ... + (2n - 1)x^{n - 1}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.21

'Translate the given text into multiple languages.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.22

'(1) (x2)(x2+x+2)(x-2)(x^{2}+x+2)\n(2) (x+1)(x+2)(x3)(x+1)(x+2)(x-3)\n(3) (x1)2(x2+2x+3)(x-1)^{2}(x^{2}+2x+3)\n(4) (x+1)(x3)(x2+2)(x+1)(x-3)(x^{2}+2)\n(5) (3x+1)(4x23x+1)(3x+1)(4x^{2}-3x+1)\n(6) (x1)(2x+1)(x2+x1)(x-1)(2x+1)(x^{2}+x-1)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.23

'Find the remainder of P(x) = x³-4x²+x-7 when x = -2'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.24

'(1) Since the solutions are \ \\alpha, \eta \, we have'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.25

'Divide P(x) by (x+1)^{2}(x-2), let the quotient be Q(x), and the remainder be R(x), then the following equation holds.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.26

'(2)\\n\\\\(\\n\\\egin{aligned}\\nS & =1+3 x+5 x^{2}+\\\\cdots \\\\cdots+(2 n-1) x^{n-1} \\\\\\nx S & =\\\\quad x+3 x^{2}+\\\\cdots \\\\cdots+(2 n-3) x^{n-1}+(2 n-1) x^{n}\\n\\end{aligned}\\n\\\\)\\n\\nSubtracting both sides,\\\\( (1-x) S=1+2\\left(x+x^{2}+\\\\cdots \\\\cdots+x^{n-1}\\right)-(2 n-1) x^{n} \\)\\n\\nTherefore, when \ x \\neq 1 \,\\\\(\\n\\\egin{aligned}\\n(1-x) S & =1+2 \\cdot \\frac{x\\left(1-x^{n-1}\\right)}{1-x}-(2 n-1) x^{n} \\\\\n& =\\frac{1-x+2\\left(x-x^{n}\\right)-(2 n-1) x^{n}(1-x)}{1-x} \\\\\n& =\\frac{1+x-(2 n+1) x^{n}+(2 n-1) x^{n+1}}{1-x}\\n\\end{aligned}\\n\\\\)\\n\\nHence \\( S=\\frac{1+x-(2 n+1) x^{n}+(2 n-1) x^{n+1}}{(1-x)^{2}} \\)\\n\\( x=1 \\text{ at } \\quad \egin{aligned}\\nS & =1+3+5+\\\\cdots \\\\cdots+(2 n-1)=\\sum_{k=1}^{n}(2 k-1) \\\\\n& =2 \\cdot \\frac{1}{2} n(n+1)-n=n^{2}\\n\\end{aligned}\\n\\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.27

'Determine the values of constants a, b, c, and d so that the equation (x + a y - 3)(2 x - 3 y + b) = 2 x^{2} + c x y - 6 y^{2} - 4 x + d y - 6 becomes an identity in terms of x and y.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.28

'\\[ 3(a x+2 b y)-(a+2 b)(x+2 y) \\]\n\\[=3 a x+6 b y-(a x+2 a y+2 b x+4 b y) \\]\n\\[=2(a x-a y-b x+b y) \\]\n\\[=2\\{ a(x-y)-b(x-y) \\} \\]\n\\[=2(a-b)(x-y) \\]\n\ a>b, x>y, thus, a-b>0, x-y>0 \\n\\[2(a-b)(x-y)>0 \\]\n\Therefore \\n\\[(a+2 b)(x+2 y)<3(a x+2 b y) \\]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.29

'Moreover, x^{3/2} + x^{-3/2} = (x^{1/2} + x^{-1/2})^3 - 3x^{1/2}x^{-1/2}(x^{1/2} + x^{-1/2})'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.30

'(A - B)(A^2 + AB + B^2)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.31

'Find the coefficient of the specified term in the following expanded expressions.(1) (2 x+3 y)^{4} [x^{2} y^{2}] (2) (3 a-2 b)^{5} [a^{2} b^{3}]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.32

'Find the general term of the expansion.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.33

'Given f(y)=y2+ya9f(y)=y^{2}+y-a-9, it can be observed that 3<\x0crac12<3-3<-\x0crac{1}{2}<3 with respect to the axis. From f(3)>0f(3)>0, we have a<3a<3 and from f(3)>0f(-3)>0, we have a<3a<-3. Therefore, we can conclude that \x0crac374<a<3-\x0crac{37}{4}<a<-3.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.34

'Practice problem: Find the coefficients of x₁^p, x₂^p, ..., xᵣ^p in the expansion of (x₁+x₂+...+xᵣ)^p.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.35

'Expand (a+b)ⁿ using the binomial theorem.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.36

'Mathematics I\n267\n\\[\egin{aligned} y_{1}+y_{2} &= \\triangle \\mathrm{OAP} - \\int_{0}^{1} (-3x^{2}+3)dx + 2y_{1} \\\\ &= \\frac{1}{2} \\cdot 1 \\cdot 3p + 3 \\int_{0}^{1} (x^{2}-1)dx + 2 \\cdot \\frac{1}{2}(2-p)^{3} \\\\ &= \\frac{3}{2}p + 3[\\frac{x^{3}}{3}-x]_{0}^{1} + (2-p)^{3} \\\\ &= \\frac{3}{2}p - 2 + (2-p)^{3} \\\\ &= -p^{3} + 6p^{2} - \\frac{21}{2}p + 6 \\end{aligned}\\]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.37

'(2) The solutions of the given equation are \ \\alpha, \eta \, therefore'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.38

'Exercise 79 volume 302 p. y = a x ^ (3) -2 x The square of the distance between the point (t, a t ^ 3-2 t) on the point and the origin is t ^ 2 + (a t ^ 3-2 t) ^ 2 = a ^ 2 t ^ 6-4 a t ^ 4 + 5 t ^ 2'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.39

'Show the conditions under which the above equation equals 0.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.40

'For a real number t, consider two points P(t, t^{2}) and Q(t+1, (t+1)^{2}).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.41

'(2) From f(a)=f(a+1) we get a^{3}-3a=(a+1)^{3}-3(a+1)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.42

'Find the coefficient of the specified term in the given expansion. (1) (x^2+2y)^5 [x^4 y^3] (2) (x^2-2/x)^6 [x^6, constant term]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.43

'A cubic equation Q(x) with a coefficient of 1 for 19x^{3} gives a remainder of -1 when divided by x-1, and a remainder of 8 when divided by x-2.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.44

'For the sequence \ \\{a_{n}\\} \ where the sum of terms from the first term to the nth term is given by \ S_{n}=2 n^{2}-n \, answer the following questions:\n1. Find the general term \ a_{n} \.\n2. Find the sum \ a_{1}+a_{3}+a_{5}+ \\ldots \\ldots+a_{2 n-1} \.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.45

'Practice book 108 page 218\n(1)\n(a)\n(√[4]{2} + √[4]{3})(√[4]{2} - √[4]{3})(√{2})'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.46

'Solve the following equations.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.47

'Check whether the following equations are identities:\n(1) (x-1)^{2}=x^{2}+1\n(2) (a+b)^{2}+(a-b)^{2}=2(a^{2}+b^{2})\n(3) \\frac{2 x+1}{2 x-1} \\times \\frac{4 x^{2}-1}{(2 x+1)^{2}}=1\n(4) \\frac{1}{3}\\left(\\frac{1}{x+1}-\\frac{1}{x+3}\\right)=\\frac{1}{(x+1)(x+3)}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.48

'General term of the sequence\nThink about the rule by which the following sequence is created, and express the \ n \th term in terms of \ n \ . Also, find the value of the 6th term.\n\\n1 \\cdot 1,-4 \\cdot 3,9 \\cdot 5,-16 \\cdot 7, \\cdots \\cdots\n\'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.49

'Expand each term on the left side of the complex equation to show that it simplifies to the simple equation on the right side.\n(2), (3) Since both the left and right sides are equally complex, transform them respectively to show that they become the same expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.50

'Factorize x⁴+2x²-8 as (x²+4)(x²-2)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.51

"Exercise 20 III \ \\Rightarrow \ Volume \ p. 471 \\n(1) Let the number of times a 1 or 2 on a large die come up be \ X \\n\n\\[ x_{n}=1 \\cdot X+(-1) \\cdot(n-X)=2 X-n \\]\n\nAs \ X \ follows a binomial distribution \\( B\\left(n, \\frac{1}{3}\\right) \\), the mean \\( E(X) \\) and variance \\( V(X) \\) of \ X \ are \\( E(X)=\\frac{n}{3} \\), \\( V(X)=n \\cdot \\frac{1}{3} \\cdot \\frac{2}{3}=\\frac{2}{9} n \\) respectively.\nTherefore, the mean \\( E\\left(x_{n}\\right) \\) and variance \\( V\\left(x_{n}\\right) \\) of \ x_{n} \ are\n\\[\egin{aligned}\nE\\left(x_{n}\\right) & =E(2 X-n)=2 E(X)-n \\\\ & =2 \\cdot \\frac{n}{3}-n=-\\frac{n}{3} \\\\nV\\left(x_{n}\\right) & =V(2 X-n)=2^{2} V(X)=\\frac{8}{9} n\n\\end{aligned}\n\\]\n(2) Since \\( V\\left(x_{n}\\right)=E\\left(x_{n}^{2}\\right)-\\left\\{E\\left(x_{n}\\right)\\right\\}^{2} \\),\n\\[E\\left(x_{n}^{2}\\right)=V\\left(x_{n}\\right)+\\left\\{E\\left(x_{n}\\right)\\right\\}^{2}=\\frac{8}{9} n+\\left(-\\frac{n}{3}\\right)^{2}=\\frac{1}{9} n(n+8)\\]\n\n(3) Since \\( S=\\pi\\left(x_{n}{ }^{2}+y_{n}{ }^{2}\\right) \\), the mean \\( E(S) \\) of \ S \ is\n\n\\[E(S)=\\pi\\left\\{E\\left(x_{n}{ }^{2}\\right)+E\\left(y_{n}{ }^{2}\\right)\\right\\}\\]\n\nNow, let's find the mean of \ y_{n} \, denoted as \\( E\\left(y_{n}\\right) \\), and its variance \\( V\\left(y_{n}\\right) \\). Let the number of times a 1 on a small die come up be \ Y \, then\n\ y_{n}=2 Y-n \\n\ Y \ also follows a binomial distribution \\( B\\left(n, \\frac{1}{6}\\right) \\), similarly as in part (1)\n\\[ \egin{array}{l}\nE\\left(y_{n}\\right)=2 \\cdot \\frac{n}{6}-n=-\\frac{2}{3} n \\\\\nV\\left(y_{n}\\right)=2^{2} \\cdot n \\cdot \\frac{1}{6} \\cdot \\frac{5}{6}=\\frac{5}{9} n\n\\end{array} \\]\nSimilarly, as in part (2)\n\\[\egin{aligned}\nE\\left(y_{n}^{2}\\right) & =\\frac{5}{9} n+\\left(-\\frac{2}{3} n\\right)^{2}=\\frac{1}{9} n(4 n+5) \\\\\n\\text {Therefore} \\quad E(S) & =\\pi\\left\\{\\frac{1}{9} n(n+8)+\\frac{1}{9} n(4 n+5)\\right\\} \\\\\n& =\\frac{1}{9} n(5 n+13) \\pi\n\\end{aligned}\\]"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.52

'In mathematics, i.e., (α-1)(β-1)(γ-1)=0, so at least one of α, β, γ is 1.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.53

'Find the remainder when the polynomial x^2020 + x^2021 is divided by the polynomial x^2 + x + 1.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.54

'(2) Let t=x+1/x, prove by mathematical induction that x^n+1/x^n will become an nth degree equation of t.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.55

'Calculate the given indices'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.56

'Let k be a real number. For the cubic equation f(x)=x^{3}-kx^{2}-1, let the three roots of the equation f(x)=0 be α, β, γ. Let g(x) be a cubic equation with a coefficient of 1 for x^{3}, and let the three roots of the equation g(x)=0 be αβ, βγ, γα.\n(1) Express g(x) in terms of α, β, γ.\n(2) Find the values of k for which the two equations f(x)=0 and g(x)=0 have a common solution.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.57

'In the practice book 8 (page 35), if the coefficient of the third term of P is denoted as a, and b, c as constants, then P = (x+1)^2(ax+b), P-4 = (x-1)^2(ax+c).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.58

'Translate the given text into multiple languages.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.59

'300'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.60

'Practice 56 (1) (First half) P_1=α+β=(1+√2)+(1-√2)=2 Also αβ=(1+√2)(1-√2)=-1 Therefore P_2=α^2+β^2=(α+β)^2-2αβ=2^2-2(-1)=6 (Second half) [1] When n=1, P_1=2, when n=2, P_2=6 Therefore, for n=1,2, P_n is an even number that is not a multiple of 4. [2] Assuming n=k, k+1, when n=k, k+1, P_n is an even number that is not a multiple of 4.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.61

'Let the first term be a, the common difference be d, and let the sum from the first term to the nth term be S_{n}. It is known that S_{5}=125 and S_{10}=500, so 1/2・5{2a+(5-1)d}=125 and 1/2・10{2a+(10-1)d}=500. Therefore, we have a+2d=25 ... (1), 2a+9d=100 ... (2). Solving equations (1) and (2) simultaneously gives a=5, d=10'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.62

'For the polynomial f(x)=x^{4}-x^{2}+1, answer the following questions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.63

'Determine the value of the real number x so that (1 + xi)(3 - i) becomes (1) a real number or (2) a purely imaginary number.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.64

'(3u-v)(-u³+3u-v) < 0'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.65

'(1) The general term of the expansion of left(x2+2yright)5\\left(x^{2}+2 y\\right)^{5} is 5mathrmCrleft(x2right)5r(2y)r=5mathrmCrcdot2rx102ryr _{5} \\mathrm{C}_{r}\\left(x^{2}\\right)^{5-r}(2 y)^{r}={ }_{5} \\mathrm{C}_{r} \\cdot 2^{r} x^{10-2 r} y^{r} . The term x4y3x^{4} y^{3} corresponds to r=3r=3, and the coefficient is'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.66

'Expand the following expressions: (a+b)³ and (a-b)³'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.67

'(1) x⁴/4 − x³ + 3x² − 2x + C (2) 2x³ − x²/2 − 12x + C (3) t³/3 − 1/2xt² − 2x²t + C'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.68

'Determine the values of constants a, b, and c, so that the equation is an identity with respect to x.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.69

'Find the coefficients of the terms a^{3} b^{2} c and a^{4} c^{2} in the expansion of (a+2b+3c)^{6}.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.70

'{P_{n}} is defined as follows.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.71

'Please list three basic functions of digital version of Chart-style reference books.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.72

'Proof of arithmetic progression'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.73

'Practice, factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.74

'In order for P to be factored as a product of linear equations in x and y, α and β must not be linear equations in y.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.75

'Expand the following expressions. (1) (a+2 b)^{7} (2) (2 x-y)^{6} (3) (2 m+n/3)^{6}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.76

'Using the binomial theorem, prove the following equation.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.77

'Because the vertex of the parabola y=x^2+bx+c lies on the line y=x, we can set the coordinates of the vertex as (k, k). Therefore, the equation of the parabola is y=(x-k)^2+k, which is y=x^2-2kx+k^2+k. The x-coordinates of the intersection points of the parabola (1) and the parabola y=-x^2+4 are the real solutions of 2x^2-2kx+k^2+k-4=0, where (1) and (2) have two distinct intersection points, so let D be the discriminant of (3) then D>0. Calculating D/4=(-k)^{2}-2(k^{2}+k-4)=-k^{2}-2k+8, therefore -k^{2}-2k+8>0, which implies k^{2}+2k-8<0, solving which gives -4<k<2. In this case, let the x-coordinates of the two intersection points be α, β (α<β), thus α and β are solutions of (3), so α+β=k and αβ=(k^{2}+k-4)/2. Hence, (β-α)^{2}=(α+β)^{2}-4αβ=k^{2}-2(k^{2}+k-4)=-k^{2}-2k+8=-(k+1)^{2}+9.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.78

'Considering the case when in mathematics B 329 n=k+2'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.79

'Please prove the identity of the function.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.80

'Transformation of Recurrence Relations, Mathematical Induction Transformation of Recurrence Relations\n- Adjacent 2 terms \\( a_{n+1} = p a_{n} + q \\(p \\neq 1) \\) For \ \\alpha \ that satisfies \ \\alpha = p \\alpha + q \\n\\[\na_{n+1} - \\alpha = p\\left(a_{n} - \\alpha\\right) \n\\]\n- Adjacent 3 terms \ p a_{n+2} + q a_{n+1} + r a_{n} = 0 \ \ p x^{2} + q x + r = 0 \ with solutions \ \\alpha, \eta \ then\n\\[\na_{n+2} - \\alpha a_{n+1} = \eta\\left(a_{n+1} - \\alpha a_{n}\\right)\n\\]\nMathematical Induction\nThe procedure to demonstrate proposition \ P \ concerning natural number \ n \ holds for all natural numbers is as follows\n[1] Prove that \ P \ is true when \ n=1 \.\n[2] Assuming that \ P \ is true for \ n=k \, prove that it is also true for \ n=k+1 \.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.81

'Assume that there exist real numbers p, q, r, s, t, u satisfying the equation x^{2}+y^{2}-5=(p x+q y+r)(s x+t y+u). When expanding the right-hand side, the coefficient of x^{2} is p s, so comparing the coefficients of x^{2} on both sides gives p s=1. Therefore, it must be the case that p is not equal to 0 and s is not equal to 0.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.82

'Let a be a real constant, and consider two circles C1: x^{2}+y^{2}=4 and C2: x^{2}-6x+y^{2}-2ay+4a+4=0'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.83

'Factorize the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.84

'Find the remainder when the polynomial P(x)=4x32x25x+3P(x)=4 x^{3}-2 x^{2}-5 x+3 is divided by the following linear expressions: (a) x+1x+1 (b) 2x12 x-1'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.85

'Use the factor theorem to factorize the following equations.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.86

'Please find the coefficients of the following expression.(6) x^6-12x^5+60x^4-160x^3+240x^2-192x+64'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.87

'Expansion 51: Factoring a quadratic 2-term expression (using the formula for roots)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.88

'Synthetic Division\nConsider the cubic polynomial P(x)=ax3+bx2+cx+d P(x)=a x^{3}+b x^{2}+c x+d divided by the linear polynomial xk x-k resulting in a quotient Q(x)=lx2+mx+n Q(x)=l x^{2}+m x+n and a remainder R R .\nThe coefficients l,m,n l, m, n of this quotient and the remainder R R can also be obtained by a method called synthetic division.\n\nProof Since the division equation P(x)=(xk)Q(x)+R P(x)=(x-k) Q(x)+R holds\n\\[\na x^{3}+b x^{2}+c x+d=(x-k)\\left(l x^{2}+m x+n\\right)+R\n\\]\nThis equation is an identity with respect to x x .\nExpanding and simplifying the right hand side\n\\[\na x^{3}+b x^{2}+c x+d=l x^{3}+(m-l k) x^{2}+(n-m k) x+(R-n k)\n\\]\nComparing coefficients on both sides\n\\na=l, \\quad b=m-l k, c=n-m k, d=R-n k\n\\]\nTherefore\n\\[\nl=a, \\quad m=b+l k, \\quad n=c+m k, \\quad R=d+n k\n\'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.89

'Find the coefficient of a term with an expansion'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.90

'Verify if the following equations are identities.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.91

'Let k be a constant. Find the value of k when the coefficient of the term a^{2}bc^{2} in the expansion of (a+kb+c)^{5} is 60. Also, find the coefficient of the term ac^{4} at this point.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.92

''

A. ...

Ask AI tutor for answer!
Join our Discord

Q.93

'Factorize the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.94

'Coefficient Determination of Identity (1)...Method of Coefficient Comparison'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.95

'[7!/(3!2!2!)]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.96

'Expand (a+b)^{4} using the binomial theorem and find the coefficients of each term.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.97

'Let {a_{n}} be the sequence: 1, 3, 8, 19, 42, 89, and let {b_{n}} be its differences. If the differences of sequence {b_{n}} form a geometric sequence,\n(1) Find the general term of sequence {b_{n}}.\n(2) Find the general term of sequence {a_{n}}. Basic Example 19'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.98

'Using the binomial theorem, find the expanded form of the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.99

'Find the coefficient of the term [x^{3} y^{2} z] in the expansion of the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.00

'To make the equation x31=a(x1)(x2)(x3)+b(x1)(x2)+c(x1)x^{3}-1=a(x-1)(x-2)(x-3)+b(x-1)(x-2)+c(x-1) an identity for all xx, determine the values of constants a,b,ca, b, c.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.01

'Determine the values of the constants a and b so that the following polynomials are divisible by the given expressions:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.02

'\ 30 \\quad \\frac{7}{3} x^{2}-x-\\frac{1}{3} \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.03

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.04

'Partial fraction decomposition'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.05

'Lesson 61: Solving Higher Degree Equations (1) - Using Factorization'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.06

'Find the coefficient of the term inside [ ] in the expanded expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.07

'In factoring polynomials of higher degree, we find the integer k that satisfies P(k) = 0, and then use the factor theorem. Here we will focus on how to find the integer k that satisfies P(k) = 0.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.08

'What is the coefficient of the expansion of (a+b+c)^{n}?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.09

'Find the coefficient of the [ ] term in the expanded expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.10

'Find the polynomials A and B that satisfy the following conditions:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.11

'Basic 45: Factorizing a quadratic equation in the realm of complex numbers'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.12

'Find the coefficient of [a b^{2} c^{2}] in the expanded form of (a+b+c)^{5}.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.13

'Binomial theorem'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.14

'Check whether the following equations are identities.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.15

'In Mathematics I, we learned about factorization and using it to solve quadratic equations. Here, we will consider methods for solving equations of degree 3 and higher using the factor theorem.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.16

'Let the two solutions of the quadratic equation (x1)(x2)+(x2)(x3)+(x3)(x1)=0(x-1)(x-2)+(x-2)(x-3)+(x-3)(x-1)=0 be alpha\\alpha and \eta\eta. Find the values of the following expressions. (1) alpha\eta\\alpha\eta (2) (1alpha)(1\eta)(1-\\alpha)(1-\eta) (3) (alpha2)(\eta2)(\\alpha-2)(\eta-2)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.17

'Transform x^2+1/(x^2-1) into 4(x^2-1)+1/(x^2-1)+4 and consider it.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.18

'Expand (a+b)^4 using the binomial theorem.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.19

'Determine the values of constants a, b, and c to make the following equations identity in terms of x: (1) \\frac{4 x+5}{(x+2)(x-1)}=\\frac{a}{x+2}+\\frac{b}{x-1} (2) \\frac{3 x+2}{x^{2}(x+1)}=\\frac{a}{x}+\\frac{b}{x^{2}}+\\frac{c}{x+1}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.20

'Given B = x^2 + x - 3, Q = 4x - 1, R = 13x - 5, find A.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.21

'(2) \\( (x-1)(x+1)(x-2)(x-3) \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.22

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.23

'The x-coordinate of the intersection points between the curve C and the line l is given by the equation x^{3}+2 x^{2}-4 x-8=0. The left side can be factored as x+2, so factoring it we get (x+2)^{2}(x-2)=0, which gives x=2,-2. Therefore, one of the x-coordinates of the points where the curve C intersects the line l, excluding the points of tangency, is 2.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.24

'Consider the sequence {a_n} from the first term to the fifth term, for n=1,2,3,4, we have a_{n+1}=a_{n}+A×10^{n}.... for all natural numbers n satisfies (1). In this case, a_{n+2}=a_{n}+B×10^{n}....(2) holds. a_{1}=11, a_{2}=101, from (2), when n is E, a_{n} is a multiple of 11, and when a_{n} is a multiple of 11, n is F.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.25

'When given A, Q, and R, how to find B?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.26

'Expand the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.27

'Factorize the following quadratic equations in the range of complex numbers:\n(1) \x^{2}-3 x-3 \\n(2) \ 2 x^{2}+4 x-1 \\n(3) \ 2 x^{2}-3 x+2 \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.28

'Let {a_{n}} be a sequence, define b_{n}=\\frac{a_{1}+a_{2}+\\cdots \\cdots+a_{n}}{n}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.29

'(3) \\( (2 x+1)\\left(3 x^{2}-x+2\\right) \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.30

'Find the coefficient of the [ ] term in the expanded expression. 6 (1) (x+y+z)^{8}[x^{2} y^{3} z^{3}] (2) (x-y-2 z)^{7} [x^{3} y^{2} z^{2}]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.31

'Prove that the equation a^{2}-bc=b^{2}-ca holds when a+b+c=0.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.32

'(1) (x-1)(x+1)(x+3)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.33

'Using the sum and product formulas'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.34

"In Mathematics I, we dealt with quadratic expressions. In Mathematics II, we will be dealing with higher degree expressions such as cubic equations. Therefore, let's first learn about expanding and factoring cubic expressions."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.35

'Find the coefficient of the term x^4 in the expansion.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.36

'Self-directed learning'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.37

'Expand and factorize a third-degree polynomial'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.38

'If the two solutions of the quadratic equation 2x²-3x+5=0 are α and β, then what is the quadratic equation with solutions α² and β²?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.39

'Factorize the quadratic equation of 512 yen using the formula for solutions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.40

'Determine the values of the constants a and b so that the following equation is an identity in terms of x:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.41

'Using the binomial theorem, find the expansion of the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.42

'Factorize the following equation: \\(x^{3}+y^{3}=(x+y)^{3}-3xy(x+y)\\).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.43

'What is the coefficient of [x^3] in the expansion of the following expressions?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.44

'How to find the general term from a recurrence relation.\\nSolve the following recurrence relations to find the general term of the sequence:\\n\\n1. Arithmetic sequence type\\n\ a_{n+1}=a_{n}+d \\\n\ [d \ is a constant \\])\\n\\n2. Geometric sequence type\\n\ a_{n+1}=r a_{n} \\\n\ [r \ is a constant \\])\\n\\n3. Difference sequence type\\n\\( a_{n+1}=a_{n}+f(n) \\)\\n\\( [ f(n) is the general term of the difference sequence \\])\\n\\nAlso,\\n\ a_{n+1}=p a_{n}+q\\\n\ p \ and \ q \ are constants, where \\( p \\neq 1, q \\neq 0 \\)\\nin the form of a recurrence relation, and find the general term of the sequence.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.45

'Basic 59: Factoring high-degree polynomials'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.46

'Find the sum of the coefficients of the terms of x^2, x^4, and x^6 in the expanded form of (1+x)(1-2x)^5.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.47

'Find the coefficient of x^{11} in the expansion of 15^4(1+x+x^2)^{8}.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.48

'Factorize the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.49

'Find the quotient and remainder when A is divided by B in each of the following cases:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.50

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.51

'Let \ \\left\\{a_{n}\\right\\}: 1,3,8,19,42,89, \\cdots \\cdots \ be a sequence. Let \ \\left\\{b_{n}\\right\\} \ be its difference sequence. When the difference sequence of \ \\left\\{b_{n}\\right\\} \ is a geometric sequence: (1) Find the general term of the sequence \ \\left\\{b_{n}\\right\\} \. (2) Find the general term of the sequence \ \\left\\{a_{n}\\right\\} \.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.52

'Find the expanded form'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.53

'Expand and factorize the given expression'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.54

'When a=2, (x-2y+1)(x+y+1), when a=-5/2, (x-2y-2)(x+y-1/2)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.55

'TRAINING 13 Find the sum of the following geometric sequences. (1) First term 4, common ratio 1/2, number of terms 7 (2) Sequence 3, -3, 3, -3, ..., number of terms n (3) Sequence 18, -6, 2, ..., number of terms n'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.56

'Find the general term of the harmonic sequence {an}, where the 2nd term is 1 and the 5th term is 1/13.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.57

"Let's find the solutions to the equation x^4 + 8x^3 + 20x^2 + 16x - 12 = 0."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.58

'Besides stirring, name two methods to increase the rate of dissolving a solid in water without changing the amount of water and solid.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.59

'【Figure 1】shows a seating chart of a classroom. There are a total of 9 seats, and all students sit facing the blackboard. To avoid students sitting next to each other in front, back, left, and right, the seats are assigned accordingly. For example, when numbering the seats, if a student sits in Seat 1, other students cannot sit in Seats 2 and 4. Answer the following questions: (1) When A, B, C, D, and E, 5 students sit, how many ways can the seats be assigned? (2) When A, B, C, D, 4 students sit, how many ways can the seats be assigned? (3) When A, B, C, 3 students sit, how many ways can the seats be assigned?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.60

'Find the sum of 1+x+x^2+⋯+x^n.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.61

'Translate the given text into multiple languages.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.62

"Reverse engineer from the 'self you want to be'."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.63

'For the sequence {an}, please answer the following questions: (1) Find the general term of the sequence {an^2 + bn^2}. Also, find lim_{n -> ∞} (an^2 + bn^2). (2) Prove that lim_{n -> ∞} an = lim_{n -> ∞} bn = 0. Also, find ∑_{n=1}^{∞} an, ∑_{n=1}^{∞} bn.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.64

'Approximate value and approximate expression'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.65

'(Example) 5) When x^2 = y^2 + 8'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.66

'Find the number of permutations that can be formed by taking any 4 letters from the word mathematics.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.67

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.68

'In all permutations of the 8 characters of PR NAGOYAJO, how many permutations contain both AA and OO, and how many permutations do not have the same characters adjacent to each other.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.69

'Factorize the following equations:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.70

'Calculate the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.71

"The number of ways to divide 4 A's, 5 B's, and 2 C's into groups is C_9^5 × C_4^2. Also, since there is no distinction between the two groups of 2 people each, the total number of ways of division is"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.72

'2 x^{2}-6 x+4'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.73

'The number of ways to choose 3 students to put in A is C_9^3'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.74

'Complete the square for 3x^2 - 5x + 1.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.75

'19 (1) \\((x+y-1)\\left(x^{2}-x y+y^{2}+x+y+1\\right)\\ (2) \\((x-2 y-z)\\left(x^{2}+4 y^{2}+z^{2}+2 x y-2 y z+z x\\right)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.76

'Simplify the like terms of the given polynomials. Also, identify the degree and constant term when focusing on the characters inside [ ].'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.77

'Expand the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.78

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.79

'(1) \\( 3(a+b)(b+c)(c+a) \\)\\n(2) \\( (a b+a+b-1)(a b-a-b-1) \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.80

'Factorize the following expression: 2x2+5xy+2y23y22x^{2}+5xy+2y^{2}-3y-2'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.81

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.82

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.83

'Factorize (x-1)(x+2)(x-3)(x+4)+24.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.84

'Factorize the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.85

'Therefore, the number of required permutations is\n\\[\n\egin{aligned}\n10080- & 24 \\times(30+30+30+20) \\\\\n& =10080-24 \\times 110=10080-2640 \\\\\n& =7440 \\text { (ways) }\n\\end{aligned}\n\\]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.86

'10\n(1)\\((x-3)(3 x-1)\\)\n(2)\\((x+1)(3 x+2)\\)\n(3)\\((a+2)(3 a-1)\\)\n(4)\\((a-3)(4 a+5)\\)\n(5)\\((2 p+3 q)(3 p-q)\\)\n(6)\\((a x-b)(b x+a)\\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.87

'Factorize the following expression.\n(1) x^{3}+3xy+y^{3}-1'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.88

'Simplify polynomial'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.89

'Calculate the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.90

'Answer the following questions about subsets of real numbers.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.91

'Factorize the following expression: 2x23xy+y2+7x5y+62x^{2}-3xy+y^{2}+7x-5y+6'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.92

'Factorize the following expressions. (1) (x+y)^{2}-4(x+y)+3 (2) 9 a^{2}-b^{2}-4 b c-4 c^{2} (3) (x+y+z)(x+3 y+z)-8 y^{2} (4) (x-y)^{3}+(y-z)^{3}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.93

'Simplify the following expressions in descending order of powers of x.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.94

'Expand the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.95

'Factorize the following expressions:\n(1) 2 x^{3}+16 y^{3}\n(2) (x+1)^{3}-27'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.96

'Expand the following expression: (4)((3 a-b)(9 a^{2}+3 a b+b^{2})).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.97

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.98

'Expand the expression (2x + 3y + z)(x + 2y + 3z)(3x + y + 2z) and find the coefficient of xyz.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.99

'What is the total number of permutations for the given string?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.00

'76 \\quad y=\\frac{1}{3}(x+1)(x-5)\n\\( \\left(y=\\frac{1}{3} x^{2}-\\frac{4}{3} x-\\frac{5}{3}\\right) \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.01

'10 \u3000 809 11 (1) \\\\ ( 2(x+2 y)(x^{2}-2 x y+4 y^{2}) \\) (2) \\\\ (x-2)(x^{2}+5 x+13) \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.02

'Please complete the square for {1}/{3}x^{2}+2x+1.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.03

'Expanding formulas'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.04

'Expanding the expression (a+b+c+d)(p+q+r)(x+y), how many terms are formed?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.05

'Calculate the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.06

'Expanding the product of polynomials can always be done by repeatedly using the distributive property, even for complex expressions. However, factorization can often lead to dead ends if calculations are carried out without considering the steps. Here, we have compiled a list of how to prioritize finding the steps for factorization. It is advisable to think about factorization while keeping these points in mind.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.07

'Given A=5x³ -2x² +3x +4 and B=3x³ -5x² +3, calculate the following: (1) A+B (2) A-B'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.08

'Please complete the square for -2 x^{2}+10 x-7.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.09

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.10

'Factorize the following equation: x2+3xy+2y2+2x+3y+1x^{2}+3xy+2y^{2}+2x+3y+1'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.11

'(3) \\((3 x+x^{3}-1)\\left(2 x^{2}-x-6\\right)\\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.12

'Simplify the following mathematical expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.13

"Intersection and union of 3 sets\nIntersection A∩B∩C is the set of all elements that belong to A, B, and C.\nUnion A∪B∪C is the set of all elements that belong to at least one of A, B, C.\nProperties of 3 sets\n(1)\n\\[\n\egin{aligned}\nn(A∪B∪C)= & n(A)+n(B)+n(C) \\\\\n& -n(A∩B)-n(B∩C)-n(C∩A)+n(A∩B∩C)\n\\end{aligned}\n\\]\n(Extension of the Principle of Inclusion-Exclusion)\n(2) \\\overline{A∪B∪C}=\\overline{A} \\cap \\overline{B} \\cap \\overline{C}, \\overline{A∩B∩C}=\\overline{A} \\cup \\overline{B} \\cup \\overline{C} \\n(Extension of De Morgan's Laws)"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.14

'Find all 28 ways of arranging 6 red beads, 2 black beads, and 1 transparent bead in a circular permutation.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.15

'Expand the expression (2x+3y+z)(x+2y+3z)(3x+y+2z) and find the coefficient of xyz.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.16

'(Example) For the equation x^2 - 2 xy + 2 y^2 = 13 (x > 0, y > 0)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.17

'Factorize the following expression:\n\nx^3 - 8y^3 - z^3 - 6xyz'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.18

'Factorize the following expressions: (1) 3x²-10x+3 (2) 3x²+5x+2 (3) 3a²+5a-2 (4) 4a²-7a-15 (5) 6p²+7pq-3q² (6) abx²+(a²-b²)x−ab'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.19

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.20

'Simplify the given expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.21

'(5) Expand the following expression: (x+y+z)(x-y-z)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.22

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.23

'Binomial theorem'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.24

'Symmetric Expression'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.25

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.26

'Transform the following equations into the form y=a(x-p)^{2}+q (complete the square).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.27

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.28

'Factorize the following equations.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.29

'12 (1) \\( (x-y)(2x+y-1) \\) (2) \\( (x+y-3)(3x+y+2) \\) (3) \\( (x+2y-1)(3x-y+2) \\) (4) \\( (x+y-z)(x-2y+z) \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.30

'A rectangle surrounded by 4 lines is formed by a combination of 2 vertical and 2 horizontal lines, so the required number is ${}_5 C_2 \\times {}_5 C_2={\\left(\\frac{5 \\cdot 4}{2 \\cdot 1}\\right)}^2=10^2=100 \\text{(units)}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.31

'How many positive integers less than or equal to 4 digits can be formed using 6 different numbers (0, 1, 2, 3, 4, 5)? Repeated use of the same number is allowed.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.32

'There are 5 separate bus routes between city A and city B. In the following cases, how many ways are there to make a round trip from city A to city B.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.33

'Assuming there are 4 white beads, 3 black beads, and 1 red bead. There are \ \\square \ ways to arrange them in a row, \ \\square \ ways to arrange them in a circle. Furthermore, there are \ \\square \ ways to thread these beads and create a loop.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.34

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.35

'Factorize the following expressions:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.36

'Answer to Exercise 1 (1) \ -x^{2}+5 x-1 \ (2) \ -3 x^{2}+3 x y-4 y^{2} \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.37

'Factorize 3x3+(9y+z)x23y(z+2y)x+2y2z -3x^{3} + (9y + z)x^{2} - 3y(z + 2y)x + 2y^{2}z as much as possible.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.38

'Calculate the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.39

'(2) 2 x^{2}-4 x+2=2(x^{2}-2 x+1)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.40

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.41

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.42

'What is the term for numbers, letters, and expressions that multiply them together?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.43

'Expansion formulas for quadratic expressions'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.44

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.45

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.46

'Expand the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.47

'Chapter 1 Calculation: Factoring with 3 factors'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.48

'Organize the following equations in descending order of powers with respect to x for (1), (2), and with respect to a for (3).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.49

'Simplify the following expressions in terms of x in descending order of powers.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.50

'Complete the square for the following quadratic equations.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.51

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.52

'Given the polynomial P=3x^3-3xy^2+x^2-y^2+ax+by.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.53

'The expansion formula is (a^2) - (b^2)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.54

'Calculate the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.55

'Expand the following expressions using the factorization formulas.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.56

'The expanding formula of (a-b)^{2} is a^{2}-2ab+b^{2}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.57

'Explain the calculation of the following expression: (a+b)^2 - (a-b)^2'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.58

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.59

"In Section 2, 'Multiplication of Polynomials,' we learned how to expand expressions in the form of polynomial products and represent them as a single polynomial. Now, let's learn the reverse process of expressing a polynomial as a monomial or a product of polynomials."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.60

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.61

'(1) \7 x^{2} + 4 x - 17\ (2) \\(x^{2}-(2 a-b) x-a\\) (3) \\(-a^{2}-2(7 b-2) a+2 b^{2}+2 b-5\\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.62

'Expand the following expression: x(x-1)(x+1)(x^2+1)(x^4+1)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.63

'The function representing the graph when it is symmetrically moved with respect to the origin of the function y=f(x) is y=-f(-x). If a and b are real numbers, and m is the minimum value of the function f(x)=x^{2}+ax+b for 0 <= x <= 1, then express m in terms of a and b.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.64

'(7) \ 4 x^{4}-13 x^{2} y^{2}+9 y^{4} \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.65

'Transform the given equation y=-x^{2}+2 x'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.66

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.67

'Expand the following expression: \n(x+2y)^2(x^2+4y^2)^2(x-2y)^2'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.68

'Please calculate the following polynomial by multiplication: (x + 2)(x - 3)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.69

'Please factorize the following polynomials.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.70

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.71

'Determine the degree and coefficient of the given monomial. Also, identify the degree and coefficient of the letters inside the square brackets.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.72

'Complete the square for the following quadratic equations'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.73

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.74

'Calculate the following expression: (4) (√3 + √5)²'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.75

'How many ways are there to select one president, one vice president, and one treasurer from 7 club members? Note that holding multiple positions simultaneously is not allowed.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.76

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.77

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.78

"Unlike before, let's consider permutations where the same item can be repeated. For example, if we take 3 characters from 2 types of characters A and B allowing duplicates, the total number of ways to arrange them in a row is 2^{3}."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.79

'Find the value of the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.80

'Organize the following expressions in descending order of powers of x. (1) x^{3}-3 x+2-2 x^{2} (2) a x-1+a+2 x^{2}+x (3) 3 x^{2}+2 x y+4 y^{2}-x-2 y+1'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.81

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.82

'Expand the following expressions: (1) (x+1)(x+2)(x+3)(x+4) (2) x(x-1)(x+3)(x+4)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.83

'(1) Expand the following expressions.(2) (3 x-1)^{3}(3) (3 x^{2}-a)(9 x^{4}+3 a x^{2}+a^{2})(4) (x-1)(x+1)(x^{2}+x+1)(x^{2}-x+1)(5) (x+2)(x+4)(x-3)(x-5)(6) (x+1)^{3}(x-1)^{3}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.84

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.85

'Factorize the following expressions. (1) (x+2)^{2}-5(x+2)-14 (2) 16(x+1)^{2}-8(x+1)+1 (3) 2(x+y)^{2}-7(x+y)+6 (4) 4x^{2}+4x+1-y^{2} (5) 25x^{2}-a^{2}+8a-16 (6) (x+y+9)^{2}-81'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.86

'Factorize the following expressions: (1) 8x³+1 (2) 64a³-125b³'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.87

'Explain the calculation of the following expression: (a-b)^2 + (b-c)^2 + (c-a)^2'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.88

'(A) -x^{2}+8x\n(1) -x^{2}+16\n(B) 2\n(I) 24'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.89

'Factorize the following expressions. (1) x^3 + 2x^2y - x^2z + xy^2 - 2xyz - y^2z (2) x^3 + 3x^2y + zx^2 + 2xy^2 + 3xyz + 2zy^2'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.90

'Transform the given expressions and find the maximum and minimum values: (1) Transform 3x^2 + 4y^2 and substitute. (2) Find the maximum and minimum values based on the range of x and y. (3) When x is a real number, transform y = (x^2 + 2x)^2 + 8(x^2 + 2x) + 10 and let t = x^2 + 2x. Find the maximum and minimum values.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.91

'In the expanded expression, the coefficient of x^5 is A, and the coefficient of x^3 is B.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.92

'Divide 10 students into several groups. In this case, how many ways are there to divide them into (1) 3 groups of 2, 3, and 5 students each. (2) 3 groups of 3, 3, and 4 students each. (3) 4 groups of 2, 2, 3, and 3 students each.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.93

'Factorize the following expressions: (1) x³-5x²-4x+20 (2) 8a³-b³+3ab(2a-b) (3) 8x³+1+6x²+3x (4) x³-9x²+27x-27'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.94

'The equation of the parabola y=x^{2}+ax+b moved symmetrically about the origin is given by replacing x and y with -x and -y respectively, resulting in -y=(-x)^{2}+a(-x)+b, which simplifies to y=-x^{2}+ax-b. Moving the parabola y=-x^{2}+ax-b horizontally by 3 units and vertically by 6 units gives the equation y-6=-(x-3)^{2}+a(x-3)-b, which further simplifies to y=-x^{2}+(a+6)x-3a-b-3. Since this matches y=-x^{2}+4x-7, we have a+6=4 and -3a-b-3=-7, solving which gives a=-2, b=10.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.95

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.96

'Permutation with same items'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.97

'Factorize the following expression: (3)(x + 1)(x + 2)(x + 3)(x + 4) - 3'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.98

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.99

'(9) Factorize the following expression: (ab)x2+(ba)xy (a-b) x^{2}+(b-a) x y '

A. ...

Ask AI tutor for answer!
Join our Discord

Q.00

'Factorize the following equations.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.01

'When expanding (a+b+c)(x+y)(p+q), how many terms will be produced?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.02

'When the parabola y=ax^{2}+bx+c is moved parallel to the x-axis by 2 units and parallel to the y-axis by -1 unit, it becomes the parabola 33y=-2x^{2}+3. Find the values of coefficients a, b, and c.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.03

'The expansion formula of (a+b)^{2} is: a^{2} + 2ab + b^{2}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.04

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.05

'Factorize the following expressions: (1) 6x^{2}+13x+6 (2) 3a^{2}-11a+6 (3) 12x^{2}+5x-2 (4) 6x^{2}-5x-4 (5) 4x^{2}-4x-15 (6) 6a^{2}+17ab+12b^{2} (7) 6x^{2}+5xy-21y^{2} (8) 12x^{2}-8xy-15y^{2} (9) 4x^{2}-3xy-27y^{2}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.06

'From 4 students, how many ways are there to select 1 chairman and 1 vice chairman? It is not allowed for the chairman and vice chairman to hold both positions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.07

'Expand the following expression: (x+1)(x+2)(x-1)(x-2)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.08

'(1) Expand the expression (x-2y+1)(x-2y-2)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.09

'If there are 3 candidates and 10 people vote anonymously, how many ways can the votes be split?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.10

'Expand the formula (x+a)(x+b).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.11

'1. (1) Degree 3, coefficient 22; aa: Degree 1, coefficient 2x22x^{2}\n2. Degree 17, coefficient \x0crac13-\x0crac{1}{3}; yy: Degree 7, coefficient \x0crac13ab7x2-\x0crac{1}{3}ab^{7}x^{2}; aa and bb: Degree 8, coefficient \x0crac13x2y7-\x0crac{1}{3}x^{2}y^{7}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.12

'(3) x^{3}+2 x^{2}-9 x-18\nx^{3}+2 x^{2}-9 x-18=(x^{3}+2 x^{2})-(9 x+18)=x^{2}(x+2)-9(x+2)=(x+2)…'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.13

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.14

'Please calculate the number of patterns for one type, three types, and four types of incense, and determine the possibilities for each scenario.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.15

'In how many ways can you divide 12 people as follows:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.16

'Calculate the following equation: (6)(4 + 2√3)(4 - 2√3)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.17

'Expand the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.18

'(9) \\ ((x+y)^{4}+(x+y)^{2}-2)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.19

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.20

'Factorize the following equations:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.21

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.22

'Explain the calculation of the following expression: (a+b)^2 + (a-b)^2'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.23

'Calculate the following expression: (5) (3√2 - √3)²'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.24

'Factorize the following expression: (4)(a-b)^{2}+c(b-a)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.25

'Symmetrically move the parabola y=x^{2}+a x+b with respect to the origin, then parallelly move 3 units in the x-axis direction and 6 units in the y-axis direction, resulting in the parabola y=-x^{2}+4 x-7. Find the values of a and b in this case.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.26

'Complete the square for the following quadratic equations.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.27

'How many ways are there to divide 12 people as follows:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.28

'Mathematics I'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.29

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.30

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.31

'Complete the square for the following quadratic equations.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.32

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.33

'When expanding (a+b+c)(x+y)(p+q), how many terms are created?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.34

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.35

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.36

'Simplify the polynomial and perform addition and subtraction.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.37

'Factorize the following equation.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.38

'Show the expansion formula of (ax+b)(cx+d).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.39

'Find the coordinates of the vertex of the parabola y=x^{2}-4 a x+4 a^{2}-4 a-3 b+9. Also, find the natural numbers a, b such that the parabola does not have any points in common with the x-axis.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.40

'Basic Example 9, 10\nExpand the following expression:\n(x+1)(x-2)(x+3)(x-4)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.41

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.42

'The product of monomials is calculated using the law of exponents. For example, 2 a b \\times 3 a^{2} b'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.43

'Problem of groupings with distinction and without distinction'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.44

'(1) y=(x1)22[y=x22x1]y=(x-1)^{2}-2[y=x^{2}-2 x-1]\\n(2) y=(x1)2+2[y=x2+2x+1]y=-(x-1)^{2}+2[y=-x^{2}+2 x+1]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.45

'Expand the following expression: (a+b+c)^2(a+b-c)^2'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.46

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.47

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.48

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.49

'(5) x^{3}+x^{2}+3 x y-27 y^{3}+9 y^{2}\nx^{3}-27 y^{3}+{x²+3 x y+9 y²}=(x-3 y)[x²+x⋅3 y+(3 y)²]+x²+…'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.50

'(1) \ x^{2}+x+\\frac{1}{4} \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.51

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.52

'Given that the coefficient of 3^{3} x^{2} is -1, the graph passes through the point (1,1), and the vertex is on the line y=x, find the quadratic function.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.53

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.54

''

A. ...

Ask AI tutor for answer!
Join our Discord

Q.55

'(2) \\( x^{3}-3 x^{2}+7=a(x-2)^{3}+b(x-2)^{2}+c(x-2)+d \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.56

'Find the value of the following expressions:\n11. (1) A=4 x^{3}-4 x^{2}+10 x-5\n(2) B=2 x^{2}-4 x+8\n'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.57

'When the expression A is divided by x+2, the quotient is B and the remainder is -5. Dividing the quotient B by x+2, the quotient is 38x^2-4 and the remainder is 2. Find the remainder when the expression A is divided by (x+2)^2. According to the conditions from Kanagawa University: A=(x+2)B-5, B=(x+2)(x^2-4)+2. Substituting (2) into (1) yields: A=(x+2){(x+2)(x^2-4)+2}-5=(x+2)^2(x^2-4)+2(x+2)-5=(x+2)^2(x^2-4)+2x-1. Therefore, when A is divided by the quadratic expression (x+2)^2, the remainder is a linear equation or a constant, hence the required remainder is 2x-1.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.58

'Find the general term a_n of the sequence {a_n} such that the sum S_n from the first term to the nth term satisfies the following relation:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.59

'Perform the factorization of the following expressions and find the coefficients:\n3. (1) (a+2 b+3 c)(a^2+4b^2+9c^2 -2ab-6bc-3ca)\n(2) (x+y-2)(x^2-xy+y^2+2x+2y+4)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.60

'Let the sum of the first to nth terms of this arithmetic progression {an} be Sn. From (1), a1 to a16 are positive numbers, and from a17 onwards, they are negative numbers; hence, Sn is maximum at n=16.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.61

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.62

'Find the sum: \\(\\sum_{l=5}^{9}(2+l^{2})\\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.63

'Factorize the following expressions: (1) a3b3b3c3a^{3} b^{3}-b^{3} c^{3}; (2) (x+y)3(y+z)3(x+y)^{3}-(y+z)^{3}; (3) 8a336a2+54a278 a^{3}-36 a^{2}+54 a-27'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.64

'Prove the following equation.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.65

'Factorize the following quadratic expressions in the range of complex numbers. (1) x^2 - 20x + 91 (2) x^2 - 4x - 3 (3) 3x^2 - 2x + 3'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.66

'(2) \ x^{4}-16 x^{2}+16 \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.67

'\\(\\left(a^{2}+b^{2}+c^{2}\\right)\\left(x^{2}+y^{2}+z^{2}\\right)=(a x+b y+c z)^{2} \\)+(a y-b x)^{2}+(b z-c y)^{2}+(c x-a z)^{2} \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.68

'Given that the polynomial f(x) divided by (x-1)^2 equals the quotient g(x) and the remainder 3x-1, and when f(x) divided by 352(x-2) the remainder is 6. Find the remainder when g(x) is divided by x-2, and the quotient of f(x) divided by (x-1)(x-2) is what x-U?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.69

'When polynomial A is divided by x+2, the quotient is B and the remainder is -5. When the quotient B is divided by x+2, the quotient is x^2-4 and the remainder is 2. Find the remainder when polynomial A is divided by (x+2)^2.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.70

'Using the expansion formula (a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}, factorize x^{3}-6x^{2}y+12xy^{2}-8y^{3}.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.71

'Expanding (x + 1)^6 using the binomial theorem gives'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.72

'Find the following sums.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.73

'Problem: Find the general term of the sequence(1):\\egin{\overlineray}{l}a_{1}=1 \\\\a_{2}=3 a_{1}-1=3 \\cdot 1-1=2 \\\\a_{3}=3 a_{2}-1=3 \\cdot 2-1=5 \\\\a_{4}=3 a_{3}-1=3 \\cdot 5-1=14 \\\\a_{5}=3 a_{4}-1=3 \\cdot 14-1=41 \\end{\overlineray}\'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.74

'The Binomial Theorem is a formula in algebra that is used to expand polynomials of the form (a+b)^n. The expansion refers to the process of multiplying out the polynomial and adding the terms to get the result.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.75

'Find the value inside the square brackets in the expansion of the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.76

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.77

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.78

'What is the condition for the linear expression x-k to be a factor of the polynomial P(x)?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.79

'(1) Since P(-2)=-3, we have P(x)=(x-1)(x+2)Q_{3}(x)+a(x+2)-3. (2) Since P(1)=4, we have 3a-3=4, therefore a=\\frac{7}{3}. Hence, the required remainder is \\frac{7}{3}(x+2)-3=\\frac{7}{3}x+\\frac{5}{3}.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.80

'(a) Factorize the following equations in the range of rational numbers:\n1) x4+2x215 x^{4}+2 x^{2}-15 \n2) 8x327 8 x^{3}-27 \n(b) Factorize the following equations in the range of real numbers:\n1) x4+2x215 x^{4}+2 x^{2}-15 \n2) 8x327 8 x^{3}-27 \n(c) Factorize the following equations in the range of complex numbers:\n1) x4+2x215 x^{4}+2 x^{2}-15 \n2) 8x327 8 x^{3}-27 '

A. ...

Ask AI tutor for answer!
Join our Discord

Q.81

'(3) \\( P(x)=\\{x(x+3)\\}\\{(x+1)(x+2)\\}-24 \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.82

'Factorize the following expressions in the ranges of (a) rational numbers, (b) real numbers, and (c) complex numbers:\n(1) x^{4}+2 x^{2}-15\n(2) 8 x^{3}-27'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.83

'[Find the coefficient of the specified term in the expanded expression]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.84

'Substituting x=-1 into x^{3}-x^{2}-5x-3 results in (-1)^{3}-(-1)^{2}-5\\cdot(-1)-3=0 \\nTherefore, x^{3}-x^{2}-5x-3 has a factor of x+1, meaning x^{3}-x^{2}-5x-3=(x+1)(x^{2}-2x-3) =(x+1)^{2}(x-3)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.85

'Find the coefficient of x^3 in the expansion of PR \\left(x^{2}-3 x+1\\right)^{10}.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.86

'Find the general term of the following sequence: -3, 2, 19, 52, 105, 182, 287, ...'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.87

'Determine the coefficient of the specified terms in the expanded expression'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.88

'(x+y)^{3} expands to x^3+3x^2y+3xy^2+y^3.\n(a-1)(a^{2}+a+1) expands to a^3 - a + a^2 - a^2 + a + 1 = a^3 + a + 1。'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.89

"Let PR be a constant. For the parabola y=x2+ax+3ay=x^{2}+a x+3-a, find the path of the vertex as 'a' varies for all real values. By transforming the equation of the parabola, we get y=left(x+fraca2right)2fraca24a+3y=\\left(x+\\frac{a}{2}\\right)^{2}-\\frac{a^{2}}{4}-a+3. Let the vertex of the parabola be P(x, y). Then x=fraca2x=-\\frac{a}{2}(1) and y=fraca24a+3y=-\\frac{a^{2}}{4}-a+3(1). From (1), we have a=2xa=-2 x. Substituting this into (2), we get y=frac14(2x)2(2x)+3=x2+2x+3y=-\\frac{1}{4}(-2 x)^{2}-(-2 x)+3=-x^{2}+2 x+3. Therefore, the required path is the parabola y=x2+2x+3y=-x^{2}+2 x+3 with the corresponding vertex coordinates left(fraca2,fraca24a+3right)\\left(-\\frac{a}{2},-\\frac{a^{2}}{4}-a+3\\right)."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.90

'Using the property that conjugate complex numbers are also solutions, when the equation f(x)=0 has an imaginary solution p+q i, then p-q i is also a solution.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.91

'Prove that at least one of x+y, y+z, z+x is 0'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.92

'Factorize the following quadratic equations in the range of complex numbers:\n(1) 15x2+14x8 15 x^{2}+14 x-8 \n(2) x22x2 x^{2}-2 x-2 \n(3) x2+2x+3 x^{2}+2 x+3 '

A. ...

Ask AI tutor for answer!
Join our Discord

Q.93

'Find the coefficients of the terms in the expanded expression'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.94

'\\( x^{4}-16 =\\left(x^{2}-4\\right)\\left(x^{2}+4\\right) =(x+2)(x-2)\\left(x^{2}+4\\right) \\)\\nTherefore, the equation is\\n\\[(x+2)(x-2)\\left(x^{2}+4\\right)=0\\]\\nSo,\ x+2=0 \ or \ x-2=0 \ or \ x^{2}+4=0 \ hence\ x= \\pm 2, \\pm 2 i \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.95

'Find the coefficient of the term a^2 b^3 c^2 in the expansion of the following expression: (a+b-2c)^7'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.96

'When S(x) is divided by (x+1)^{2}(x-3), the quotient is denoted by Q_{1}(x).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.97

'(1) Find the coefficient of [x^9] in the expansion of (2x^3 - 3x)^5'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.98

'When the constant is k, find the value of k when the quadratic equation x2+3xy+2y23x5y+kx^{2}+3xy+2y^{2}-3x-5y+k can be factored into the product of linear equations in x,yx,y. Also, find the result of the factored form. '

A. ...

Ask AI tutor for answer!
Join our Discord

Q.99

'Let the sequence \ \\left\\{a_{n}\\right\\} \ be an arithmetic sequence with first term 1 and common difference 3. Define \ S_{n} \ as the sum of products of two distinct terms from the first \ n \ terms of the sequence \ \\left\\{a_{n}\\right\\} \. For example, \ S_{3}=a_{1} a_{2}+a_{1} a_{3}+a_{2} a_{3} \. Find \ S_{10} \.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.00

'Find the coefficients of the specified terms in the following expansion expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.01

'Find the solution to the equation (x + 1)(x + 3) = x(9 - 2x).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.02

'There are 5 cards with numbers from 1 to 5 written on them. When 2 cards are drawn from them simultaneously, find the expected value of the numbers on the drawn cards (E(5 X^{2}+3)) and the variance (V(3 X + 1)).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.03

'Expand (a + b)^n using the binomial theorem.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.04

'(1) Let a, b be constants. Suppose the polynomial in x, x^{3}+ax+b is divisible by (x+1)^{2}. Find the values of a and b. (2) Let n be a natural number greater than 2. Determine the values of the constants a, b when the polynomial x^{n}+ax+b is divisible by (x-1)^{2}.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.05

'Factorize the following polynomial: x^3 - 6x^2 + 11x - 6.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.06

'Expand (x+1/x^2+1)^5 and find the term that does not contain x.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.07

'(1) (Equation) \\( = \\frac{x^{2}-1}{x+1} = \\frac{(x+1)(x-1)}{x+1} \\)\ = x-1 \ (2) And Equation \=\\frac{x^{2}}{x^{2}-1}-\\frac{2 x}{x^{2}-1}+\\frac{1}{x^{2}-1} \\\(=\\frac{x^{2}-2 x+1}{x^{2}-1}=\\frac{(x-1)^{2}}{(x+1)(x-1)}=\\frac{x-1}{x+1} \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.08

'Find the coefficient of x³ in the expansion of (1) [Aichi Institute of Technology]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.09

'What is the condition for the linear expression ax+b to be a factor of the polynomial P(x)?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.10

'To ensure that the first expression is divisible by the second expression, determine the values of the constants a, b, c, d, e.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.11

'In the formula Sn=fracaleft(rn1right)r1 S_n = \\frac{a\\left(r^n-1\\right)}{r-1} , with a=1,r=2,n=10 a=-1, r=2, n=10 , the sum is calculated to be\n\\[ \\frac{-1 \\cdot\\left(2^{10}-1\\right)}{2-1} = -(1024-1) = -1023 \\]\nIn the formula Sn=na S_n = na , with n=10,a=3 n=10, a=3 , the sum is calculated to be\n\ 10 \\cdot 3 = 30 \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.12

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.13

'The given equation is as follows:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.14

'Find the values of s, t, p, q for which the function f(x)=a x^{3}+b x^{2}+c x+d satisfies \\[\\int_{-3}^{3} f(x) d x=s \\cdot f(p)+t \\cdot f(q)\\]. Also, ensure that p ≤ q.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.15

'Expand the following cubic equations and factorize: (a+b)^{3}, (a-b)^{3}, (a+b)(a^{2}-ab+b^{2}), (a-b)(a^{2}+ab+b^{2})'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.16

'Factorize the following expressions:\n2. (1) (a) Product: (3 x-y)(9 x^{2}+3 x y+y^{2})\n(b) Product: 9(a+2 b)(a^{2}-2 a b+4 b^{2})\n(c) Product: (2 x-y z)(4 x^{2}+2 x y z+y^{2} z^{2})\n(2) (x+4)^{3}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.17

'Prove the following equations hold.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.18

'(1) \\((x-7)(x-13)\\)\n(2) \\((x-2-\\sqrt{7})(x-2+\\sqrt{7})\\)\n(3) \\(3\\left(x-\\frac{1+2 \\sqrt{2} i}{3}\\right)\\left(x-\\frac{1-2 \\sqrt{2} i}{3}\\right)\\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.19

'Let a be a negative constant. Find the maximum and minimum values of the function f(x)=2x³-3(a+1)x²+6a x on the interval -2 ≤ x ≤ 2.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.20

'Fill in the blank to make the equation hold: (x-1)^{3}-7(x-1)^{2}+17(x-1)-9 = 314'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.21

'Find a_n using the following steps: (1) a_{n}=2+\\frac{3}{n+2} (2) a_{n}=\\frac{3 \\cdot 5^{n}+1}{5^{n}-1}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.22

'Calculate the following expressions: (1) \\( \\frac{1}{(x-3)(x-1)}+\\frac{1}{(x-1)(x+1)}+\\frac{1}{(x+1)(x+3)} \\) (2) \ \\frac{1}{a^{2}-a}+\\frac{1}{a^{2}+a}+\\frac{1}{a^{2}+3 a+2} \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.23

'When k is a constant, find the value of k such that the quadratic equation x^2+3xy+2y^2-3x-5y+k can be factored into the product of linear equations in x and y. Also find the result of the factoring in that case.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.24

'(1) \ a_{n}=5^{n+1}-20 \\cdot 3^{n-1} \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.25

'Determine the values of the constants a, b, c such that the following equation holds for x and y: (1) x^2 + axy + by^2 = (cx + y)(x - 4y)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.26

'Coefficient of the term x y^2 z in (2)(2x-\\frac{1}{2}y+z)^4'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.27

'(1) (a) \\( \\left(x^{2}-3\\right)\\left(x^{2}+5\\right) \\)\n(b) \\( (x+\\sqrt{3})(x-\\sqrt{3})\\left(x^{2}+5\\right) \\)\n(c) \\( (x+\\sqrt{3})(x-\\sqrt{3}) \\) \\( \\times(x+\\sqrt{5} i)(x-\\sqrt{5} i) \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.28

'(1) \\( (x+y)^{3}=x^{3}+3x^{2}y+3xy^{2}+y^{3} \\) (2) \\( (a-1)(a^{2}+a+1)=a^{3}-1^{3}=a^{3}-1 \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.29

'Find the sum of the arithmetic sequence: \\( \\sum_{k=5}^{14}(2k-9) \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.30

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.31

'Chapter 1 Formulas and Proofs 27 Find the coefficient of x^4 when expanding (2+x)^6, and find the coefficient of x^3 when expanding (1+x)^6(2+x)^6. [Kwansei Gakuin University]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.32

'For the shape B_{n+1}, focus on the rightmost column. As shown in Figure 1, when tiles are placed vertically in the protruding part, the remaining part matches shape A_{n+1}. In this case, there are a_{n+1} ways to tile. On the other hand, as shown in Figure 2, when tiles are placed horizontally in the protruding part, there are 3 ways to do so, and the remaining part matches shape B_{n}. In this case, there are b_{n} ways to tile. Therefore, b_{n+1} = a_{n+1} + b_{n}, so b_{2} = a_{2} + b_{1} = 11 + 4 = 15'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.33

'Find the coefficients of the following expressions:\n1. (1) x^{3}-x^{2}+\x0crac{1}{3} x-\x0crac{1}{27}\n(2) -8 s^{3}+12 s^{2} t-6 s t^{2}+t^{3}\n(3) 27 x^{3}+8 y^{3}\n(4) -a^{3}+27 b^{3}\n(5) 64 x^{6}-48 x^{4} y^{2}+12 x^{2} y^{4}-y^{6}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.34

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.35

'Rewrite the right-hand side as'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.36

'Let g(x) = (x^3 - 2x^2 - 45x - 40) / (x - 8), find the decimal part of g(2020). Here, the decimal part of a real number a is defined as a - n where n is the greatest integer less than or equal to a.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.37

'Find the general term of the following arithmetic sequences.\n(a) 1, -\\frac{1}{2}, -2, -\\frac{7}{2},\n(b) p+1, 4, -p+7, -2p+10,\n(2) In an arithmetic sequence where the 9th term is 26 and the 18th term is 53, what term is 134 in the sequence? Also, which term is the first to exceed 1000.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.38

'Find the general term of an arithmetic sequence. Let the first term be a and the common difference be d.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.39

'Please factorize the following expressions. (1) 2(x1)211(x1)+152(x-1)^{2}-11(x-1)+15 (2) x2y2+4y4x^{2}-y^{2}+4y-4 (3) x410x2+9x^{4}-10x^{2}+9 (4) (x2+3x)22(x2+3x)8(x^{2}+3x)^{2}-2(x^{2}+3x)-8'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.40

'Expand the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.41

'Expand the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.42

'The given expression (x+1)(x+2)(x+3)(x+4) - 24 factors into (x^2 + 5x + 4)(x^2 + 5x + 6)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.43

'Expand (x+y+1)(x^{2}-xy+y^{2}-x-y+1).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.44

'(2) \\[\egin{aligned}(x-1)(x-2)(x+1)(x+2) & =(x-1)(x+1) \\times(x-2)(x+2) =(x^{2}-1) \\times(x^{2}-4) =(x^{2})^{2}-5 x^{2}+4 =x^{4}-5 x^{2}+4 \\]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.45

'Translate the given text into multiple languages.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.46

'\\[ (4) \\left(-2 a x^{3} y\\right)^{2}\\left(-3 a b^{2} x y^{3}\\right) =(-2)^{2} a^{2}\\left(x^{3}\\right)^{2} y^{2} \\times(-3) a b^{2} x y^{3} =4 a^{2} x^{6} y^{2} \\times(-3) a b^{2} x y^{3} =4 \\cdot(-3) a^{2+1} b^{2} x^{6+1} y^{2+3} =-12 a^{3} b^{2} x^{7} y^{5} \\]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.47

'Expand the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.48

'For real numbers x and y satisfying |2x+y|+|2x-y|=4, the possible range of 2x^2+xy-y^2 is 11 ≤ 2x^2+xy-y^2 ≤ 9.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.49

'Translate the given text into multiple languages.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.50

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.51

'Factorize the following equation.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.52

'Let a be a real number and b be a positive constant. Find the minimum value m of the function f(x)=x^{2}+2(a x+b|x|). Furthermore, as the value of a changes, plot a versus m graph for m.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.53

'Find the equation of a parabola that is moved parallelly by -1 in the x-direction and 2 in the y-direction such that it is transformed onto the parabola y = x^2 + 3x + 4.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.54

'Factorize the following expressions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.55

'(1) Factorize the left side as'

A. ...

Ask AI tutor for answer!