# Tagged Questions

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, checking for irreducibility. This tag is rarely used as the only tag for a question.

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### Finding coefficients of two polynomials

Let $n$ be a natural number. Let $f(x)=\prod_{i=-n}^{n}(x-i)$. If $k$ is an even integer, then the coefficient of $x^k$ is zero. The coefficient of $x^{2n-1}$ is ...
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### Roots of a quartic polynomial

If $a, 3a, 5a, b, b + 3,$ and $b+5$ are all roots of a fourth-degree polynomial equation where $0<a<b$, compute all possible values of $a$. By the fundamental theorem of algebra, the polynomial ...
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### A stronger form of Rolle's Theorem in the direction of number of roots of $f'(x)$

Today I read an interesting generalization of the Rolle's Theorem for Polynomials in $E. 28$ of E. J. Barbeau's book on Polynomials. It says that if $a, b$ are two consecutive zeroes of polynomial ...
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### Solutions of an equation of degree $n>4$

I know that the Abell-Ruffini theorem prove that we cannot solve a general equation of degree $n>4$ with radicals. But I've read that quintic equations can be solved by means of elliptic modular ...
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### Getting K heads out of N biased coins problem (formula generation ).

Problem- Given a set of coins $n$ with each coin $i$ having $P_i$ probability to give heads. Find the probability of getting $k$ heads, when all coins are tossed together. Hi I have solved this ...
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### $p,q$ coprime polynomials - are $p^n,q^m$ coprime? [on hold]

Suppose that $p,q \in K[x]$ are coprime (there is no polynomial that divides both) and let $n,m \in\mathbb{N}$. Are $p^n$ and $q^n$ coprime and, if so, how to prove it?
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### A variation of Buchberger algorithm

Let $I$ be an ideal of a polynomial ring $R$. Fix a monomial order. Denote the $S$-polynomial of $f, g\in R$ by $S(f, g)$ and denote the gcd of their leading terms by $T(f, g)$. Consider the ...
Let $I$ be an ideal of a polynomial ring $R=k[x_1,\ldots,x_n]$ over a field $k$. A Groebner basis of $I$ is a finite generating set $\{g_1,\ldots,g_m\}$ such that every leading monomial (according to ...
### Conics and conics of the form $ax^2+by^2+c=0$
The problem of finding rational points on conics is usually discussed (for example in the book of Silverman and Tate) for conics of the form $ax^2+by^2+c=0$. I assume that those conics are in ...