# Tagged Questions

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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, $P_a(t) = t^2 -2t +1 -a$ ...
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### How can I find the roots of a quartic equation, knowing one of its roots?

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?
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### Find the intersections of the functions

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 ...
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### Polynomial Divison of unknowns

How to complete this $$P(X)=\frac{2x^4-7x^3+5x^2+ax+b }{ 2x^2+x-1}$$ so that the division is without a remainder? When it is divided it gives two equation.
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### $f(x) = x^3 - x$ then $f(n)$ is multiple of 3

If $f(x) = x^3 - x$ then $f(n)$ is multiple of 3 for all integer $n$. First i tried $$f(n) = n^3-n=n(n+1)(n-1)\qquad\forall n\ .$$ When $x$ is an integer then at least one factor on the right is ...
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### Understanding 2012 AMC 12B #23

Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23$, $-21$, $-17$, and $-15$, and $Q(P(x))$ has zeros at $x=-59$,$-57$,$-51$ and $-49$. What is ...
For all ordered triples $(p,q,r)$ define the polynomial $$f_{p,q,r}(x)=x^3-px^2+qx-r$$ Let $a_{1},a_{2},a_{3},b_{1},b_{2},b_{3},c_{1},c_{2},c_{3}$ be (not necessarily distinct) positive reals such ...