# Questions tagged [polynomials]

This tag is used 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.

17,651 questions
906 views

### Solving quartic equations

Given the following quartic equation: $$x^4-2x^3-7x^2+8x+12=0$$ Could anyone give some techniques required to solve any quartic equation (apart from this one) if they exist?
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### Galois group of $(x^3-2)(x^5-1)$ over $\mathbb{Q}$.

I am studying for my Galois theory final for tomorrow (and I'm really getting burned out), I need help with the following question: Galois group $G$ of $f(x)=(x^3-2)(x^5-1)$ over $\mathbb{Q}$. Let ...
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### A question on polynomial (Divide a polynomial function)

When a polynomial $f(x)$ is divided by $(x-2),$ the remainder is $7$. When $f(x)$ is divided by $(x+1)$ the remainder is $-2$. (a) If the remainder is $px+q$ when $f(x)$ is divided by $(x-2)(x+1)$, ...
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### Let $m,n\in \mathbb{Z}$ and $p(x)=x^3+mx+n$ be such that if $107\mid p(x)-p(y)\implies 107\mid x-y$. Prove that $107\mid m$.

Let $m,n\in \mathbb{Z}$ and $p(x)=x^3+mx+n$ be such that for an integers $x,y$ we have: $$107\mid p(x)-p(y)\implies 107\mid x-y$$ Prove that $107\mid m$. I'm not sure what to do here. I can only ...
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### If the polynomial f(x) with real coefficients is non-negative for every $x \in \mathbb{R}$ then all its real zeros have even multiplicity

I'm dealing with this problem that says that if $f(x)$ with real coefficients is non-negative for every $x \in \mathbb{R}$ then there exists the polynomials $f_1(x)$ and $f_2(x)$ from $\mathbb{R}[x]$ ...
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### Prove that $5x^2−2xy−8x+ 2y^2−2y+ 5 \ge 0$ for all $x, y\in\mathbb R$. When does equality occur?

I tried grouping the $x$'s and the $y$'s but that didn't get me anywhere. I know that $5x^2, 2y^2$, and $5$ are always positive. I am not sure what to try next.
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### Sum of sixth power of roots of $x^3-x-1=0$

Question: Find the sum of sixth power of roots of the equation $x^3-x-1=0$ My First approach: Let $S_i$ denote the sum of $i^{th}$ power of roots of the given equation. Now, multiplying given ...
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### How to split a quartic into two quadratics?

I have a quartic in $\Bbb Z[x]$ with very large coefficients that I know splits into two quadratics in $\Bbb Z[x]$. What is the best way to do find the quadratics?
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### Assumption $d>2$ on Proposition 2.12 from Knapp's Elliptic Curves

I'm going through Knapp's book on elliptic curves and I got stuck in a minor detail. This is a part of the proof of Proposition 2.12: I could understand everything except for this little detail: ...
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### Determine derivative of polynomial from graph on a multiple choice question [on hold]

On a multiple choice question I need to determine which of the following graphs: Multiple choice graphs When derivated, yields: The graph Is there a general, fastest way to solve this kind of ...
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### Is this proof of a irreducibility criterion in an integral domain correct?

This is an exercise from Grillet's "Abstract Algebra" (page $145$, proposition $10.10$). Let $R$ be an integral domain, let $I$ be an ideal of $R$, and let $\pi\colon R\to R/I$ be a canonical ...
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### What are the origins of the Routh-Hurwitz Criterion?

The Routh-Hurwitz criterion and method are usually taught in a cookbook format. Essentially, you follow a recipe for placing the coefficients into a table and perform "figure 8" multiplication and ...
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### If $f(x)$ is a polynomial, it cannot represent primes only [on hold]

If I take $u = f(x)$ and $v = f(x + ku)$ where $k$ is any integer and then if somehow I can prove that $u$ is a factor of v but I don't have any idea about the next step.
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### Can roots be $y$-intercepts for the quadratic function

I know this must be stupid question, but I was wondering why cannot a quadratic or any polynomial equation be in format of $$x=ay^2+by+c$$ and to find roots we set $x=0$. In short, can the $y$ ...
V = $\Bbb R_3 [x]$ and W,U $\subseteq$ V are sub vector spaces. U=$span${$1 - x, x^2, x^2-x^3, -1+x-x^2+2x^3$} W={p(x)$\in\Bbb R_3 [x]$ | p(1)=0 ^ p(2)+p(0)=0} Find basis to W, U+W, U$\cap$W