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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Loomis and Sternberg Chapter 2, problem 2.19: defining the degree of a polynomial on a vector space (over R)
Exercise 2.19 of chapter 2 of L&S is:
A polynomial on a vector space V is a real-valued function on V which can be
represented as a finite sum of finite products of linear functionals. Define the ...
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Sum of 100th power of roots of a biquadratic equation [closed]
If $a,b,c$ and $d$ are the roots of a polynomial $x^4 -2x^3 +4x^2 -8x +16$ ,
Then find $a^{100} +b^{100} +c^{100} +d^{100}$
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Where does the third solution come from?
There's a well known trick with polynomials. If we have
\begin{align*}
&&(x-r_1)(x-r_2) &= 0 \\
&& x^2 -(r_1+r_2)x + r_1r_2 &=0 \tag{*}\label{*} \\
&\text{so}& x^2 &...
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Degree with which a polynomial changes with some small change
Soft question: I was curious as to how one could measure the degree with which a polynomial is perturbed. More formally, let $P(x) \in \mathbb{C}$ be a polynomial and $\epsilon$ be a very small number,...
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Prove that $x^6+5x^2+8$ is reducible over Z (integer)?
$attempts:-$
1] I tried to replace $X^2=t$ but nothing click after that .
2] then I tried to replace this polynomial say P(x) by P(x+1) or P(x-1) to apply Eisenstein's Irreducibility Criterion Theorem ...
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Action of symmetric group on polynomial ring
In Example 1.1 of Eisenbud's Commutative Algebra he writes that the symmetric group $\Sigma = S_r$ acts on the polynomial ring $S = k[x_1, \dots, x_r]$ by
$$\sigma(f)(x_1, \dots, x_r) = f(x_{\sigma^{-...
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Confusion on proof of a polynomial of degree n is $\Omega (x^n)$
This is a solution from Rosen's Discrete Math textbook for the problem of proving a polynomial of degree n is of order $x^n$, that is $\Theta(x^n)$.
I understand the proof for the polynomial of degree ...
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Real roots of $x^4+ax^3+bx^2+cx+1=0,$ when $a,b,c$ are real and $b\ge\frac{a^2+c^2}{4}$
For real $a,b,c$ and
$$b \ge \frac{a^2+c^2}{4}\tag{*}$$
the given polynomial equation
$$f(x)=x^4+ax^3+bx^2+cx+1=0\tag{**}$$
can be re-written as
$$f(x)=(x^2+ax/2)^2+(b-a^2/4-c^2/4)x^2+(cx/2+1)^2\ge 0\...
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How do I prove this statement about polynomials of degree greater than 1? [closed]
Consider the polynomial equation $p(x)=a$. Let $x_0$ be a solution. I want to prove that, if $p(X)$ and $a$ have the same sign for some $X\in\mathbb R$, then
$$\frac{a}{p(X)}>1\implies \frac a{p(X)}...
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Integrate $\int f(x) sn(x,m) dx$ where $f(x)$ is a polynomial
To find the antiderivative of a product of a polynomial $f(x)$ and a sine function we can use this formula:
$$\int f(x) sin(x) dx = F'(x) sin(x) - F(x) cos(x) + C $$
where $F(x) = f(x) - f''(x) + f^{(...
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Does there exists something like the BKK Theorem for polynomials over finite fields?
I'm trying to count the number of common $\mathbb{F}_q^\times$-zeros of some polynomials $f,g \in \mathbb{F}_q[x,y]$. I thought I had a solution using the BKK theorem but I reread the theorem ...
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Proving existence of a root in the unit disk using Rouché's Theorem
Let $a\in\mathbb{C}$, and let $n \ge 2$. Prove that the polynomial $2022+az+2023z^n$ has a root in the unit disk, $D(0,1)$.
There's an algebraic way to solve this with Vieta's formulas, by observing ...
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Evaluation of Polynomial Over Finite Prime Field [duplicate]
Let $P$ be a polynomial with coefficients in $\mathbb{Z}$. Let $q$ be a prime. Let $S$ be a set such that $P(x) \in \mathbb{N}, 0 < P(x) < q$ for any $x \in S$.
The question is, can we evaluate $...
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Characterizing all total derivatives of a homogeneous differential polynomial
A total derivative here means a differential polynomial (D.P.) $P(u,u_1,u_2,...,u_n)$ (Note: $u=u(x),u_k=\partial_x^ku(x)$) such that there exists another D.P. $G$ such that $P=\partial G$.
The ...
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When $x=\frac{u(f_i,g_j)}{v(f_i,g_j)}$ implies $x=\frac{u(f_i(x,0),g_j(x,0))}{v(f_i(x,0),g_j(x,0))}$ ($x=\frac{xy}{y}$ does not imply $x=\frac{0}{0}$)
Let $f_i=f_i(x,y), g_j=g_j(x,y) \in \mathbb{C}[x,y]$,
$1 \leq i \leq n$, $1 \leq j \leq m$, be such that
$f_i(x,0) \neq 0$ and $g_j(x,0)=0$.
Assume that $\mathbb{C}(f_1,\ldots,f_n,g_1,\ldots,g_m)=\...
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