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## 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\'

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'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]'

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

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

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'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 \'

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'(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'

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'(2) The general term of the expansion is $\\frac{10!}{p!q!r!} \\cdot 1^{p} \\cdot x^{q} \\cdot\\left(x^{2}\\right)^{r}=\\frac{10!}{p!q!r!} \\cdot x^{q+2 r}$ where $p+q+r=10 \\quad\\cdots\\cdots (1), p \\geqq 0, q \\geqq 0, r \\geqq 0$.\nThe term with $x^{4}$ occurs when $q+2 r=4$, that is when $q=4-2r$.'

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

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'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'

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'Translate the given text into multiple languages.'

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

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'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.'

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

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'Translate the given text into multiple languages.'

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'(1) $(x-2)(x^{2}+x+2)$\n(2) $(x+1)(x+2)(x-3)$\n(3) $(x-1)^{2}(x^{2}+2x+3)$\n(4) $(x+1)(x-3)(x^{2}+2)$\n(5) $(3x+1)(4x^{2}-3x+1)$\n(6) $(x-1)(2x+1)(x^{2}+x-1)$'

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

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

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'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.'

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'(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\\)'

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'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.'

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'\\[ 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) \\]'

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'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})'

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'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}]'

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

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

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'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}\\]'

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

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'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'

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

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

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

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'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]'

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'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.'

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'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} \.'

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

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'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)}'

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'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\'

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'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.'

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"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}\\]"

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

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

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

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'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.'

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'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).'

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'Translate the given text into multiple languages.'

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'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.'

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'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'

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

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'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.'

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

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

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

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

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

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

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'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.'

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

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

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'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.'

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

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'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 \.'

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'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.'

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'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'

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

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

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

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

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'Synthetic Division\nConsider the cubic polynomial $P(x)=a x^{3}+b x^{2}+c x+d$ divided by the linear polynomial $x-k$ resulting in a quotient $Q(x)=l x^{2}+m x+n$ and a remainder $R$.\nThe coefficients $l, m, n$ of this quotient and the remainder $R$ can also be obtained by a method called synthetic division.\n\nProof Since the division equation $P(x)=(x-k) Q(x)+R$ 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 $x$.\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\'

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'Find the coefficient of a term with an expansion'

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'Verify if the following equations are identities.'

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'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.'

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

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Join our Discord#### Q.96

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

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'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'

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

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

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

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

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

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

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'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.'

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

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

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

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

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

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'Check whether the following equations are identities.'

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'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.'

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

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

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'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}'

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

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'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.'

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'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.'

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'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 \'

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

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'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}]'

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

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"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."

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

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'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?'

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

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

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

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

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

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'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.'

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'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.'

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

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

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'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\\} \.'

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

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'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'

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

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

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

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'【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?'

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'Translate the given text into multiple languages.'

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

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'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.'

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

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'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.'

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"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"

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

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'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)'

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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 [ ].'

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

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

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'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\\]'

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'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)\\)'

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

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Join our Discord#### Q.90

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

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'Factorize the following expression: $2x^{2}-3xy+y^{2}+7x-5y+6$'

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'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}'

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

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

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

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

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

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'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) \\)'

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'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) \\)'

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

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

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'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.'

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

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

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

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"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)"

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

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

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

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

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'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'

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

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

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'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) \\)'

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'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)}'

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'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.'

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'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.'

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'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.'

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

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

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

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

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Join our Discord#### Q.49

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

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'Complete the square for the following quadratic equations.'

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

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'Expand the following expressions using the factorization formulas.'

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

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

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"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."

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'(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\\)'

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

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'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.'

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

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

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

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'Complete the square for the following quadratic equations'

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

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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.'

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"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}."

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'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'

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

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'(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}'

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'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'

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

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

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'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'

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'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.'

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

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'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.'

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'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'

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'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.'

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

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'(9) Factorize the following expression: $(a-b) x^{2}+(b-a) x y$'

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

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'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.'

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

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'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}'

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'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.'

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

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

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'1. (1) Degree 3, coefficient $2$; $a$: Degree 1, coefficient $2x^{2}$\n2. Degree 17, coefficient $-\x0crac{1}{3}$; $y$: Degree 7, coefficient $-\x0crac{1}{3}ab^{7}x^{2}$; $a$ and $b$: Degree 8, coefficient $-\x0crac{1}{3}x^{2}y^{7}$'

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'(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)…'

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

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

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

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

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

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

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'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.'

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'Complete the square for the following quadratic equations.'

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

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'Complete the square for the following quadratic equations.'

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

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'Simplify the polynomial and perform addition and subtraction.'

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'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.'

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

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

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'Problem of groupings with distinction and without distinction'

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'(1) $y=(x-1)^{2}-2[y=x^{2}-2 x-1]$\\n(2) $y=-(x-1)^{2}+2[y=-x^{2}+2 x+1]$'

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

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'(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+…'

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'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.'

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

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'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'

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'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.'

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'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:'

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'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)'

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'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.'

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

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'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'

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'\\(\\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} \\)'

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'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?'

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'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.'

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'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}.'

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

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'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}\'

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'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.'

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

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

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'(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}.'

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

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

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'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'

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

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'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)'

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

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

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

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'(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。'

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

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'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.'

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

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

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'\\( 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 \'

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

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

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

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

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'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} \.'

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

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

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'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)).'

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'(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}.'

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

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

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'(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} \\)'

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

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

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

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

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'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.'

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'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})'

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'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}'

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'(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)\\)'

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'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.'

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

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'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}'

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'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} \'

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'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.'

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'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)'

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

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'(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) \\)'

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'(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 \\)'

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

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'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]'

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'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'

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'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}'

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'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.'

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'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.'

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

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'Please factorize the following expressions. (1) $2(x-1)^{2}-11(x-1)+15$ (2) $x^{2}-y^{2}+4y-4$ (3) $x^{4}-10x^{2}+9$ (4) $(x^{2}+3x)^{2}-2(x^{2}+3x)-8$'

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

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'(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 \\]'

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'Translate the given text into multiple languages.'

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'\\[ (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} \\]'

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'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.'

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'Translate the given text into multiple languages.'

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'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.'

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'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.'