# 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.

17,848 questions
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### Determine polynomial function of degree 4.

The graph of a polynomial function $f(x)$ of degree $4$ with real coefficients has a local maximum at $(-3|3)$ and a local minimum at $(1|0)$ and no other local extremal points. Determine the function....
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### Order Polynomial of Matrix Commutative Property

Given a matrix $A$ and a polynomial $P$ as its linear factors $$\prod_{i=1}^{n}(\lambda_i-x)$$ For $P(A)$, does it matter in which order the linear factors are multiplied, seeing as matrix ...
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### Kernel of endomorphism for polynomials from $f(x)$ to $f(2x+1)$

I am looking for a kernel of a map, producing $f(2x+1)$ out of $f(x)$. where $f(x)$ is an arbitrary polynomial of degree $n$. I thought of trying to write this transformation as a matrix and then to ...
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### Prove that polynomial doesn't admit a particular solution

Prove that the equation$$x^4+(a-2)x^3+(a^2-2a+4)x^2-x+1=0$$ does not admit $$x=-2$$ as a triple root.
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### Homomorphisms from $k[x,y]$ to $k[x,x^{-1},y]$

Let $k$ be a field of characteristic zero and let $f$ be a $k$-algebra homomorphism from $k[x,y]$ to $k[x,x^{-1},y]$. Denote $u:=f(x)$ and $v:=f(y)$. Assume that $u=x+\tilde{u}$ and $v=y+\tilde{v}$, ...
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### Why is “division by $(z-1)$” valid here?

Is there an easy way to justify: $$x(x-1)(x+1) \equiv x(x^2-1) \Rightarrow (x-1)(x+1) \equiv x^2-1,$$ even for $x=0$? I seemingly have to divide by $x$ which should place the restriction $x \neq 0$ on ...
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