# Questions tagged [polynomials]

For both basic and advanced questions on polynomials in any number of variables, including, but not limited to solving for roots, factoring, and checking for irreducibility.

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### Finding if roots of a polynomial function are shifted when adding another polynomial function?

Not sure if this question has an easy answer but I was wondering if there is a heuristic or some theorem that helps proving which way the roots are being shifted for an equation $f(x)=g(x)+h(x)$. For ...
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### Condition to have the convexity of the infimum of multi variate polynomial

I consider $f\in\mathbb{R}[x,u]$ a polynomial function where $x\in\mathbb{R}^n$ and $u\in\mathbb{R}^m$. I am interested in $$g(x)=\inf_{u\in U}\{ f(x,u)\}$$ Where $U$ is a compact set. I am trying ...
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### If $P(x)$ is a fifth degree polynomial such that $P(x)+1$ is dividable by $(x-1)^3$ and $P(x)-1$ is dividable by $(x+1)^3$, How to find $P(x)$

I saw this question: If $P(x)$ is a fifth degree polynomial such that $P(x)+1$ is dividable by $(x-1)^3$ and $P(x)-1$ is dividable by $(x+1)^3$,find $P(x)$. I tried my best to find such function but ...
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### Are there any problems about the difference between set theoretic definitions of polynomials?

I am a novice about this question, so if there is a misunderstanding then I apologize for it. As for Peano axioms, if I choose Zermelo natural numbers, and you choose von Neumann ones, then this doesn'...
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### $f$ is irreducible if the polynomial reduced $p$ is irreducible and the degrees are the same

Let $f$ be an irreducible polynomial and $h(f)$ the polynomial with coefficients reduced modulo a prime $p$. Then if $\deg(f)=\deg(h(f))$ and $h(f)$ is irreducible then $f$ is irreducible as an ...
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### Probability of choosing a constant polynomial [closed]

Suppose we have a polynomial $$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$$ where all the coefficients are whole numbers which includes Zero. Suppose we now randomly choose the values of these coefficients ...
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### Convexity of cubic affine and quartic affine?

I am a post graduate student who is currently study the subject of convex optimization. From my class I know that affine function is convex and also $x^r$ for $r>=1$ is convex for non-negative $x$. ...
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### How to factor $(x-y)^5 + (y-z)^5 + (z-x)^5$

We see that polynomial is cyclic. For $x=y$, $P(y)=0$ so $x-y$ factors polynomial. Because it is cyclic we instantly know other two factor $(y-z)$ and $(z-x)$. $P(x,y,z)=(x-y)(y-z)(z-x)*N(x)$ Because ...
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### How to factor $k^2(k+1)^2 -c(k+1)^2 +k^2c +c^2 +c$

Basically I need to prove that $k^2(k+1)^2 -c(k+1)^2 +k^2c +c^2 +c = [k(k+1) -c]^2$. When I first saw polynomial I thought that it could’t be factored into anything meaningful, let alone a square of ...
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### What's the maximum and minimum value of $P(-2)$?

Consider a quartic polynomial $$P(x) = ax^4+bx^3+cx^2+dx+e$$ Where $a, b, c, d, e\in \{0, 1, 2\}$ and $0, 1, 2$ are chosen at least once. What's the maximum and minimum value of $P(-2)$? If we were to ...
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### Bounding a ratio of parametrized sums, uniformly

Let $\gamma_{p}\ge0$ be a sequence such that $\sum_{p\ge2}\gamma_p^2p^k<\infty$ for every $k\in\mathbb{N}$. Define: $C_k(s)=\sum_{p\ge2}\gamma_p^2s^pp^k$ for $s\in\left(0,1\right]$. I want to show ...
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### How to solve the roots of the following function numerically?

I have the following function $$f(n) = \frac{\sqrt{n}+ \sqrt{n+ 16 (\sqrt{n}+1)}}{8} - \frac{1}{2}\left(\sqrt[3]{n + \sqrt{n^2 - \frac{1}{27}}}+ \sqrt[3]{n - \sqrt{n^2 - \frac{1}{27}}}\right).$$ It is ...
1 vote
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### Integrality of a quotient

Consider two positive integers $m$ and $n$ with $m>n$. I would like to prove that the quotient $$\prod_{k=0}^{n-1}\dfrac{X^{2^m-2^k}-1}{X^{2^n-2^k}-1}$$ is a polynomial in fact. What I did is to ...
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