# Tagged Questions

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### Find all complex and real roots of higher degree polynomials, given one root

$2+3i$ is a zero of $f(x)=x^4-4x^3+17x^2-16x+52$, find all of the zeros of $f(x)$ thanks!
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### Finding a cubic equation from the relation between the roots

I'm trying to solve this problem: $x^3 - x^2 - 3x -10 = 0$ has roots α,β,γ. Let u = −α+β+γ. Show that u+2α=1, and hence find a cubic equation having roots −α+β+γ, α−β+γ, α+β−γ. I was able to ...
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### Solve $x+3y=4y^3,y+3z=4z^3 ,z+3x=4x^3$ in reals

Find answers of this system of equations in reals$$\left\{ \begin{array}{c} x+3y=4y^3 \\ y+3z=4z^3 \\ z+3x=4x^3 \end{array} \right.$$ Things O have done: summing these 3 equations give ...
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### How to expand this polynomial division?

My Physics teacher gave me a problem and its solution, what I have todo is to expand the solution, but when I do it I do not get to the same solution he says is the right one, here is the problem: ...
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### Solve $x+\frac{2}{y}=3,y+\frac{2}{z}=3,z+\frac{2}{x}=3$ in reals

find answers of this system of equations in real numbers$$\left\{ \begin{array}{c} x+\frac{2}{y}=3 \\ y+\frac{2}{z}=3 \\ z+\frac{2}{x}=3 \end{array} \right.$$ Things i have done: first i ...
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### Prove $\frac{a}{(b-c)^2}+\frac{b}{(c-a)^2}+\frac{c}{(a-b)^2}=0$ if $\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0$

if $a,b,c$ are real numbers and $$\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0$$ Prove $$\frac{a}{(b-c)^2}+\frac{b}{(c-a)^2}+\frac{c}{(a-b)^2}=0$$ things i have done: using the assumption i ...
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### Solve $x^2+y^2=4, z^2+t^2=9, xt+yz=6$ in integers

find answers of this system of equations in integers$$\left\{ \begin{array}{c} x^2+y^2=4 \\ z^2+t^2=9 \\ xt+yz=6 \end{array} \right.$$ things I have done: we can observe that ...
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### Prove $a^4+b^4+(a-b)^4=c^4+d^4+(c-d)^4$ if $a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$

if $a,b,c,d$ are positive real numbers and $$a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$$ Prove $a^4+b^4+(a-b)^4=c^4+d^4+(c-d)^4$ Things i have done: from assumption $a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$ I ...
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### Factoring $a^4(b-c)+b^4(c-a)+c^4(a-b)$

I was solving the question that wanted to factor $a^4(b-c)+b^4(c-a)+c^4(a-b)$. My idea was to factor a $(c-a)$ in first step.So $$b(a^4-c^4)+ac(c^3-a^3)+b^4(c-a)=a^4(b-c)+b^4(c-a)+c^4(a-b)$$ ...
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### Prove $a^2+b^2+c^2=\frac{6}{5}$ if $a+b+c=0$ and $a^3+b^3+c^3=a^5+b^5+c^5$

if $a,b,c$ are real numbers that $a\neq0,b\neq0,c\neq0$ and $a+b+c=0$ and $$a^3+b^3+c^3=a^5+b^5+c^5$$ Prove that $a^2+b^2+c^2=\frac{6}{5}$. Things I have done: $a+b+c=0$ So ...
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### Factoring the following polynomials

Factorize the following polynomial: $t^3 -9t +8$ $t^6 -91t^2 +90$
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### Solving a system of polynomial equations in three variables ($x^2-yz=18$, $y^2-zx=8$, $z^2-xy=-7$)

I am looking for a way to solve the following system of polynomial equations in three variables: \begin{align*} x^2-yz&=18\\ y^2-zx&=8\\ z^2-xy&=-7 \end{align*} I've tried ...
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### Find integral solutions for $2x^2+y^2=2\times(1007)^2+1$

Find integral solutions to the equation $$2x^2+y^2=2\times(1007)^2+1$$ I tried: I rewrote the equation as $2x^2+y^2=2028099$. I found that $y_{max}=1424$ and $y$ must be odd, so I set ...
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### Find polynomials $f (x)$, $g(x)$, and $h(x)$

In an elementary Algebra book (101 problems in Algebra) there was a question I solved but when I looked at the solutions I didn't get it. it says find Polynomials $f(x)$, $g(x)$, $h(x)$ such that for ...
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### how can I find equation variables?

I have the following equations : $$\begin{cases}K = \frac{B – 3}{20}\\ K = (20S+3)R+S\\ K = 20S^2 + (20N+7)S + N\\ N=S-R \end{cases}$$ - And I have the $B$ values, e.g : 173, 283, 2343, 834343 ...
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### How to solve a nonlinear system of three equations involving rational functions?

How do I get $a$, $b$, and $c$? Given $$X=\frac{a+\frac{1}2b}{a+b+c}$$ $$Y=\frac{b(\frac{\sqrt3}{2})}{a+b+c}$$ $$Z=\frac{76a+150b+29c}{255}$$ in other words How do i get $a$, $b$, and $c$ on the ...
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### Determine the nature of $f(x)$

Consider a polynomial $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients. Determine and prove the nature of $f(x)$. Can someone ...
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### Solve in $\mathbb{R}$: $(x^2-3x-2)^2-3(x^2-3x-2)-2-x=0$

Solve in $\mathbb{R}$: $(x^2-3x-2)^2-3(x^2-3x-2)-2-x=0$ I'm supposed to solve this equation. It's from a math contest so solving it by hand would be preferable (no quartic formulas). I thought ...
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### generalized way of finding pair solutions of an equation

I want to find out pair solutions of this equation: $$x^{2}-79y^{2}=1$$ This is a hyperbola equation. I sketched its graph, but that didn't help me. I think the square from (form?) of $x$ and $y$ is ...
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### For $5$ distinct integers $a_i$, $1\le i\le5$, $f(a_i)=2$. Find an integer b (if it exists) such that f(b) = $9$.

Here's an interesting question I came across.The person who gave it to me told me that it should not take more than $3$ minutes to solve this question. But I could not find any definite solution :( ...
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### How to find the roots of $-x^3+3x^2-7x+5 = 0$?

I would like to understand how to go about solving something like this, not just get the solution but some kind of methodology (that hopefully makes as much intuitive sense as possible); I honestly ...
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### Nice polynomial reducibility: $x^n+4$

Problem: Find all $n\in \mathbb{N}$ such that $f(x)=x^n+4$ is reducible in $\mathbb{Z}[x]$. It seems $n=4k$ is the only one (the factorization follows easily from Sophie Germain's identity in this ...
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### Solve the following equation: $\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$

Solve the following equation: $$\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$$ I know it's from a Math Olympiad but I don't know which and I couldn't find it on the internet. Expanding everything ...
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### Polynomial of degree 5 divisible by other polynomials.

I need some help with a problem. Find a polynomial $f(x)$ of degree $5$ such that both of these properties hold: $f(x)-1$ is divisible by $(x-1)^3$. $f(x)$ is divisible by $x^3$. I can't seem to ...
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### Polynomial $f(x)$ degree problem.

Suppose the polynomial $f(x)$ is of degree $3$ and satisfies $f(3)=2$, $f(4)=4$, $f(5)=-3$, and $f(6)=8$. Determine the value of $f(0)$. How would I solve this problem? It seems quite complicated... ...
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### Roots Of Monic Cubic

I'm currently preparing for the USA Mathematical Talent Search competition. I've been brushing up my proof-writing skills for several weeks now, but one area that I have not been formally taught about ...
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### Polynomial division problem

Let $f(x) = x^{10}+5x^9-8x^8+7x^7-x^6-12x^5+4x^4-8x^3+12x^2-5x-5.$ Without using long division (which would be horribly nasty!), find the remainder when $f(x)$ is divided by $x^2-1$. I'm not sure ...
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### Not understanding steps in Algebraic simplification

The simplification in question is that the expression goes from $(4-x)(6-x)(3-x)-8(3-x)=0$, to $(3-x)(8-x)(2-x)=0$ I don't understand how one goes from the first expression to the second. I ...
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### How can this equality be established by elementary algebraic means?

Let $x \geq 1$. Then is it true that $2x^3 - 3x^2 + 2 \geq 1$? If so, how can I show this using only elementary ideas such as factorisation? Of course, I can demonstrate this using the methods of ...
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### Factor Cyclic Polynomial

Factor $(a+b)(b+c)(c+a)+abc$. I know this is a cyclic polynomial, but I don't know how to solve problems like this. What should I do?
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### Find the value of $\frac{S_{5}S_{2}}{S_{7}}$

If $a$, $b$, $c$ $\in \mathbb R$, we define $S_{k}=\frac{a^k+b^k+c^k}{k}$ (where $k$ is a non-negative integer). Given that $S_{1}=0$, find the value of $$\frac{S_{5}S_{2}}{S_{7}}$$ I tried: ...
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### Solving the complex polynomial

For the complex polynomial $z^3 -5z^2 +(7-2i)z +6i-3 = 0$ $1)$ show that $2+i$ is a root. $2)$ solve the given equation. Attemp to solve: I'm not really sure how to solve this, but I ...
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### give a complete factored form of the polynomial $-6a^5+48a^4+12a$

Give a complete factored form of the polynomial $-6a^5+48a^4+12a$ I have tried solving this equation and I just cant figure it out. Help me, and give me the answer.
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### Relationship between constant term and roots

Does anyone know of a relationship between the constant term of a polynomial and the roots of the polynomial? Specifically, if we know the constant term, is it possible for a root which divides the ...
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### How to find a polynomial with $f(1), f(4),f(9)$ prime and coefficients in $\{1,2,3…10\}$?

How to find a polynomial with $f(1), f(4),f(9)$ prime and coefficients in $\{1,2,3...10\}$? I can't think of any way on how to generate such types of polynomials? Also, would they have a minimum ...
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### Formula alteration

is there any way to transform the formula$\frac {1-x}{x-3}$ into something that can be easily sketched, or which will help eliminate $x$ from the denominator?
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### AMM Polynomial equation

Solve the equation: $x^7+7px^5+14p^2x^3+7p^3x+q=0$ I've tried obvious things like factorization or maybe guessing a solution. I'd appreciate a solution not too far from high school level.
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### Find the maximum value of $\sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}$

If $x\in\mathbb{R}$ find the maximum value of $$\sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}$$ I tried this: Let $$y= \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}$$ For maxima ...
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### solving the inequalty

are there any ways to solve :$x^4 -6x^3 +28x^2 -64x +96 >0$ ?
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### Irreducibility of some polynomial

Let $p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1$, for some $k \geq 2$ fixed. I would like to know if $p(x)$ is irreducible in $\mathbb{Q}[x]$.
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### How “separable” (not in that sense) is a polynomial?

Since "separable" is used for different meaning in separable polynomial and separation of variable, I am having trouble searching for anything related to my question. So I hope someone can help with ...
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### Find the roots of the equation $(1+xi)^n+(1-xi)^n=0$

Find the roots of the equation $f(x)=(1+xi)^n+(1-xi)^n=0$. I'm having problems finding the roots...this is what I've done: First I expressed $(1+xi)^n$ and $(1-xi)^n$ in trigonometric form and ...
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### Given roots (real and complex), find the polynomial

This is not a duplicate of theory of equations finding roots from given polynomial. Given that the roots (both real and complex) of a polynomial are $\frac{2}{3}$, $-1$, $3+\sqrt2i$, and $3+\sqrt2i$, ...
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### Graphing polynomials

Sketch a graph of the polynomial $P(x)=(x-2)^2(x+1)^3$. You must plot and label the x and y intercepts and these should be the only points you plot. How do I sketch the graph of a polynomial?
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### How to establish these two facts about polynomials?

Let $f(x) := \sum_{k=0}^n c_k x^k$ be a polynomial of degree $n\geq 0$ with real coefficeints such that $f(x) = 0$ for $n+1$ distinct real values of $x$. Then how to prove that each $c_k = 0$ and ...
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### Roots of $x^2 +2x +2$ Over $\mathbb{C}$

Find the roots of $x^2 +2x +2$ over $\mathbb{C}$ I need to prove somehow that the roots will be $(1 + i) , (1 - i)$ Any ideas how can I find those roots in a simple way?
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### Using Remainder or Factor Theorems to Find Coefficient

I'm studying the remainder and factor theorems and a question asks: -4 is a root of $x^4 + ax^3 - 19x^2 - 46x + 120$ What is the value of a? Since -4 is a root then I can deduce that x+4 is a ...
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### Explanation on characterstic polynomial

$A_2 = \begin{pmatrix} 1 & 1 \\ a & 1 \end{pmatrix}$ So the characteristic polynomial of $A_2$ is $P_a(t) = (t-1)^2 - a$ Then, $P_a(t) = t^2 -2t +1 -a$ ...
I need to decompose (in $\Bbb{C}[x]$) the function $$f(x) = x^4 + 4x^3 - 4x^2 + 24x + 15$$ in its simplest form, knowing that $1 - 2i$ is one of its roots. Any ideas?
I have $f(x)=-x^2+4$ a parabola and $g(x)=\sqrt{(4-x^2})$ a semi circle with a raduis of $2$ if I say $g(x)=f(x)$ and solve for $x$. I should find the points at which $x$ intercepts ...