# 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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### Question about polynomial ring and coefficients

Let $R=k[x_0,...,x_n]$ be the polynomial ring in $n+1$ variables and let $F=c_1f_1+ \cdots +c_kf_k \in R$ with $c_i \in k$. Is it possible to multiply $F$ with some element from $k$ such that the ...
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### If $\alpha$ is a root of $f(t) = t^n + a_{n-1}t^{n-1} + \cdots + a_0$, then $|\alpha| \leq n \max_i |a_i|$

Let $f(t) = t^n + a_{n-1}t^{n-1} + \cdots + a_0$. Let $\alpha$ be a root of $f$. Then show that $\alpha \leq n \max_{i} |a_i|$. I could only figure it out for the special case where $|a_i| < 1$ ...
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### Can a polynomial equation always be manipulated to give a recurrence formula?

Let $p(x)$ be a real (or maybe complex) polynomial. Suppose we wish to (numerically) solve $p(x) = 0$. This can be done for example with Newton's method of course, but I was thinking about if you "...
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### Find $p$, such that $\frac1{20} = (1 - p)^{19}p$

I need help to solve for $p$, where $p$ is a probability, i.e. it lies in the interval $[0,1]$. $$\frac1{20} = (1 - p)^{19}p.$$ How would one solve for $p$? Thnx
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### polynomial with nonzero coefficients at prime degree terms

Let $P(x)$ be a polynomial with integer coefficients. Show that there is a non-zero polynomial $Q(x)$ with integer coefficients, such that the product $$P(x)Q(x)=\sum_{k\ge 0}a_k x^k$$ has only ...
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### Graphing: Given two points on a graph, find the logarithmic function that passes through both.

Is there such a method to do this? I would like to come up with a logarithmic function (a graph that looks like a square root graph) that passes through two given points. Haven't had any luck in ...
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### $5x\big(1+\frac{1}{x^2 +y^2}\big)=12$ ; $5y\big(1-\frac{1}{x^2 +y^2}\big)=4$ find $x$ and $y$
I already tried to solve using substitution and cross multiplication method . I got the first simplified (1)$$\frac{12}{5x}=1+ \frac{1}{x^2} +y^2$$ (2) $$\frac{4}{5y}=1- \frac{1}{x^2+y^2}$$ Adding (...
Consider the following quartic equation: $$x^4 + rx^3 + r^2x^2 + r^3x + r^4 - 1 = 0$$ By Lodovico Ferrari solution, this equation must possess four radical solution provided that $r$ is a rational ...