Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

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3
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1answer
153 views

Factoring polynomials of the form $1+x+\cdots +x^{p-1}$ in finite field

Suppose $p$ and $q$ primes and $p$ is odd. Then, are there nice and clever ways to factorize polynomials of the form $$f(x)=1+x+\cdots +x^{p-1}$$ in the ring $\mathbb{F}_q[x]$ ? What about the case ...
4
votes
1answer
184 views

$x^4-3x^3-9x^2+2=0$, why does wolframalpha give complex solutions when they are real?

Although $x^4-3x^3-9x^2+2=y$ intersects with the $x$-axis $4$ times (this is shown in the graph) Wolframalpha gives me complex solutions. Why does this happen? Thanks.
2
votes
3answers
1k views

Express $\cos 6\theta $ in terms of $\cos \theta$

I think I'm supposed to use the chebyshev polynomials, as in $$ \cos n \theta = T_n(x) = \cos(n \arccos x)$$ But no idea what now?
3
votes
1answer
67 views

root of an equation

I have the following equation: $$\sum_{k=0}^n \frac{a_k}{a_k+x}=1$$ where all the $a_k$'s are positive real numbers. For $n=2$ the roots are $x={}_{-}^+\sqrt{a_1a_2}$, but for $n\geq 3$ the ...
0
votes
1answer
494 views

Factor $x^6 +5x^3 +8$

I wanted to know, how can I factor $x^6 +5x^3 +8$, I have no idea. Is there any method to know if a polynomial is factored. Just some advice will do. Help appreciated. Thanks.
0
votes
1answer
342 views

How to find rational solutions to this 4th order polynomial?

I am interested in the congruent number problem which involves finding a rational solution to $y^2=x^3-x n^2$ This equation is currently unsolved and many have tried. This equation gives the X and ...
7
votes
4answers
235 views

How do we show that an ideal of polynomials is prime

I'm trying to solve this exercise: To do so, I'm trying to prove that $(X_1^2+X_2^2+X_3^2)$ is a prime ideal. Suppose now $f,g\in \mathbb R[X_1,X_2,X_3]$ and $f\cdot g\in (X_1^2+X_2^2+X_3^2)$, i. ...
2
votes
1answer
69 views

How can I get polynomial $p(x)$?

$p(x)$ is divided evenly into $x^{2}+1$, and $p(x)+1$ is divided evenly into $x^{3}+x^{2}+1$. How can I get $p(x)$?
3
votes
5answers
208 views

Polynomials - Solutions

How I can find the exact solutions of this polynomial? I can not get to the exact roots of the polynomial ... what methods occupy for this "problem"? $$x^3+3x^2-7x+1=0$$ Thanks for your help.
0
votes
2answers
111 views

Non-negative polynomials decomposition

I'm trying to understand whether the following is true: I have the polynomial $P(x)$ which is non-negative when $-1<x<1$ Is it correct that $P(x)$ could be represented by a sum with ...
0
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1answer
38 views

Conversion of $F(x) = {1 \choose 0} (2t^2 - 3t + 1) + {3 \choose 1} (4t - 4t^2) + {1 \choose 2} (2t^2 - t)$

I have a textbook with a calculation step that is pretty unclear to me: $$F(x) = {1 \choose 0} (2t^2 - 3t + 1) + {3 \choose 1} (4t - 4t^2) + {1 \choose 2} (2t^2 - t)$$ $$= {-8 \choose 0} t^2 + ...
0
votes
1answer
119 views

Problem: when the sum of two squares is a square

Please, I need help to solve the following problem: Let $K$ be a field with characteristic different from $2$ and $3$. Show that the following statement are equivalent: The sum of two ...
1
vote
1answer
106 views

Prove that there not real roots with $P(x)=ax^3+bx^2+cx+d, $

let $P(x)=ax^3+bx^2+cx+d,a,b,c,d\in R$, such that $$\min{\{d,b+d\}}>\max{\{|c|,|a+c|\}}$$ show that $P(x)=0$ have no real roots in $[-1,1]$
6
votes
4answers
306 views

Polynomials Question: Proving $a=b=c$.

Question: Let $P_1(x)=ax^2-bx-c, P_2(x)=bx^2-cx-a \text{ and } P_3=cx^2-ax-b$ , where $a,b,c$ are non zero reals. There exists a real $\alpha$ such that ...
1
vote
1answer
116 views

Proper map, what's wrong?

"A map $f$ from $\mathbb R^2$ to $\mathbb R^2$ is proper if the full preimage of every compact set under $f$ is compact. Prove that every complex polynomial $f$ regarded as a self-map of the plane of ...
0
votes
1answer
72 views

number of solutions in homogeneous system

What is the maximum possible number of solutions of homogeneous system $N \times N$ ($N$ variables, $N$ equations) of degree $2$, where in each equation we have linear terms in $x_i$ and quadratic ...
0
votes
1answer
14k views

How to use polyfit() function in Matlab?

First of all i am new to Matlab, and not sure whether to use polyfit or something else for my problem. I plotted a graph with the following matlab code: ...
3
votes
2answers
155 views

How can I solve this system of non-linear equations?

I'm trying to solve this system of equations: $$\left\{ \begin{array}{lcr} a+c & = & 0\\ b+ac+d & = & 6\\ bc+ad & = & -5\\ bd & = & 6 \end{array} \right.$$ The book ...
5
votes
2answers
211 views

Seemingly simple system of equations

I have the following system: $x^{2} + y = 31$ $x + y^{2} = 41$ As I try to solve it via simple substitution, I get into 4-th power equations, which I can simplify to $(x-5)(x^{3}+5x^{2}-37x-184)$ ...
21
votes
5answers
713 views

Solving a peculiar system of equations

I have the following system of equations where the $m$'s are known but $a, b, c, x, y, z$ are unknown. How does one go about solving this system? All the usual linear algebra tricks I know don't apply ...
3
votes
2answers
447 views

The system of equations $x^2 + y^2 - x - 2y = 0$ and $x + 2y = c$

I have $(1.) \quad x^2 + y^2 - x - 2y = 0 \\ (2.) \quad x + 2y = c$ Solving for $y$ in $(2.)$ gives $y = (c - x) / 2$ Is there a way to simplify equation $(1.)$? Because at the end I arrive at ...
0
votes
1answer
211 views

Solving simultaneous equations with 3 parts

I keep trying to solve this problem, but i keep on getting crazy answers, i think i am right up to a certain point and then doing something wrong, the question is to solvie this : $$ \begin{align*} ...
1
vote
1answer
67 views

Polynomial whose only values are squares

Given a polynomial $ P \in \Bbb Z [X] $ such that, $ P (x)$ is the square of an integer for all integers x, is $ P $ necessarily of the form $ P (x)= Q (x)^2$ with $ Q \in \Bbb Z [X]$?
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vote
2answers
386 views

Test to see if a degree $\leq4$ polynomial is factorable

I'm in the middle of a programming project and we'd like to have tests to determine if polynomials in $\mathbb{Z}[x]$ of degrees up to 4 are factorable over $\mathbb{Q}$. A test that computes the ...
3
votes
1answer
31 views

Prove that $q(a_i)\in \{a_1,…, a_n\}$

Let $p(x)$ and $q(x)$ be polynomials with rational coefficients such that $p(x)$ is irreducible over $\mathbb{Q}$. Let $a_1,..., a_n\in \mathbb{C}$ be the complex roots of $p$, and suppose that ...
0
votes
1answer
98 views

Cyclotomic Polynomials over $\mathbb Q$ and reduction modulo $p$.

Let $p$ be prime, and let $\pi : \mathbb Z \to \mathbb Z / (p\mathbb Z)$ be the canonical projection $\pi(z) = z + p\mathbb Z$. Define its extension $\pi : \mathbb Z[x] \to \mathbb Z/(p\mathbb Z)[x]$ ...
2
votes
1answer
817 views

Difference between polynomial functions and polynomials and why these two polynomial functions are equal?

Here's an excerpt from abstract algebra book that I'm reading and my question is given later: The difference between a polynomial and a polynomial function is mainly a difference of viewpoint. Given ...
-2
votes
4answers
95 views

Can x- ( $ \frac x 2 + \frac x 4$ ) be a polynomial?

I was confused whether it is a polynomial or not because most of the polynomials which I see don't have brackets like this.
2
votes
1answer
121 views

Proof $d(x)=\text{GCD}(f(x),g(x))$

Question1: Proof: If $d(x)|f(x),d(x)|g(x)$, $d(x)$ is an combination of $f(x)$ and $g(x)$, then $d(x)$ is $\text{GCD}(f(x),g(x))$. My proof 1 $d(x)=u(x)f(x)+v(x)g(x)$. ...
-2
votes
2answers
54 views

Can $(\frac{1}{4} t^{2} + \frac{1}{2} t +200) – \frac{2}{3} t$ be a polynomial?

I know that the expression inside the brackets is a polynomial but when the expression outside the brackets is combined with it, can it be considered a polynomial?
1
vote
1answer
31 views

Can a known change in both axes of a polynomial function be used to find the value of the independent variable?

I have a polynomial function of one variable, $d = f(t)$. I have a known change in t which causes a known change in $d$. What I want to find is the total $t$. Is there a way of solving this other than ...
0
votes
1answer
82 views

What is the semantic of square brackets after the set denoting coefficients of polynomial?

I have the following excerpt: Unless stated otherwise, we assume all polynomials take integer coefficients, i.e. a polynomial $f \in \mathbb{Z}[{\bf y}, x]$ is of the form $$f(y, x) = a_m · x^{d_m} ...
0
votes
1answer
90 views

Help with non-linear system of equations

This system of equations $$\begin{align} xy+yz+zx & =3 \\ \\ x^4+y^4+z^4 & =3\end{align}$$ How to solve this system of equations? Any help, Plz. Thank all
1
vote
2answers
101 views

Factoring any single-variable polynomial in $\mathbb C$

The fundamental theorem of algebra says $$ \forall p(x):\mathbb C \to \mathbb C,\ p(x) = a\prod_{n=0}^m\big(b_nx+c_n\big) $$ where $p(x)$ is a single-variable polynomial, and $\{a;m\}\cup\{\forall ...
3
votes
1answer
88 views

Does every connected component of $\{z : |P(z)|<1 \}$ contain a zero?

Let $P(z)$ be a complex non-constant polynomial. Let $G$ be a connected component of open set $\{z : |p(z)|<1 \}$. How to prove that $G$ contains a zero of $P$? I have no idea how to even ...
1
vote
0answers
67 views

Simplify $\frac{[m+n-1]!}{[m]![n]!}$ where $[k]=x^k-x^{-k}$ and $[k]!=[2][3]…[k]$.

Adopting the notation $[k] = x^k - x^{-k} $ and $[k]! = [2][3]...[k]$ (note that $[1]$ is omitted), and letting $m,n$ be two integers greater than $1$ such that $n>m$ and $gcd(m,n)=1$, would it be ...
0
votes
1answer
60 views

how do you find the highest common factor of two multivariate polynomials?

How do you find the highest common factor of two multivariate polynomials? I am happy to get answers that are only useful for polynomials over the real numbers, as that is what I am dealing with.
4
votes
1answer
258 views

Showing that a root $x_0$ of a polynomial is bounded by $|x_0|<(n+1)\cdot c_{\rm max}/c_1$

I have doubts about the following problem (Problem 3.21 from Sipser's "Introduction to the Theory of Computation"): Let $c_1 x^n + c_2 x^{n-1} + \cdots + c_n x + c_{n+1}$ be a polynomial with a ...
0
votes
1answer
84 views

Prove fact about polynomial in uncountable fields

$F$-uncountable field. $I_{i}$-ideal in $F[x_{1},...,x_{n}]$ $F^{n}=\cup_{i=1}^{\infty}V(I_{i})$   $V(I_{i})\subseteq V(I_{i+1})$ Prove that $\exists k, V(I_{k})=F^{n}$ All that I've find is that ...
10
votes
2answers
271 views

Prove $\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k}$ and more

The current issue (vol. 120, no. 6) of the American Mathematical Monthly has a proof by probabilistic means that $$\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k} $$ for ...
2
votes
1answer
63 views

What does an expression $[x^n](1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}(1-x^4)^{-1}…$ mean?

I came across the function that describes number of partitions of $n$ (I mean partitions like $5=4+1=3+2=3+1+1$ and so on. There was defined a Cartesian product: ...
3
votes
1answer
95 views

Help with generating functions.

Background. Let $P_0(y)=2y-3$ and define recursively $$P_{n+1}(y)=4y\cdot P_n'(y)+(5-4y)\cdot P_n(y).$$ I would like to know as many properties of $P_n$ as I can. For example, it can be shown that ...
2
votes
2answers
96 views

Characterization of polynomial injection from Q to Q?

I want to know if we can find (or characterize) all the polynomials $f(x) \in \mathbb{Q}[x]$ that induces an injection $f : \mathbb{Q} \rightarrow \mathbb{Q}$ by evaluation. Some examples are $x, ...
1
vote
0answers
78 views

prove that polynomial has root of unity

Prove that $ f=x^n\pm x^m\pm1 $ is either irreducible over rationals or has a root which is a of unity. I tried to see what if $x=|r|e^{i\phi}$ but I have no proper result.
3
votes
4answers
342 views

Factorize $x^3-3x+2$

How can I factorize $x^3-3x+2$ ? The answer that I got on the internet is $(x-1)^2(x+2)$. It would be nice if anyone could also tell what these type of equations are called and where can I learn ...
0
votes
3answers
121 views

Galois group $\operatorname{Gal}(f/\mathbb{Q})$ of the polynomial $f(x)=(x^2+3)(x^2-2)$

Find the Galois group $\operatorname{Gal}(f/\mathbb{Q})$ of the polynomial $f(x)=(x^2+3)(x^2-2)$. Any explanations during the demonstration, will be appreciated. Thanks!
0
votes
1answer
75 views

What is the pattern of this sequence?

I went though this pattern and I think the results might be interesting. It was a long one but I'm only showing the first five (to make things look simpler). $$0,1,a+b,a^2 + b^2 + \frac 32ab , ...
0
votes
0answers
106 views

How to find the nearest power product?

We call power products the integers of the form $x^m*y^n$ for $m$, $n$, $x$, $y \in \mathbb{N}$. Given a number $u \in \mathbb{N}$, find the closest power product. How does one solve this ...
1
vote
1answer
2k views

multiplication in GF(256) (AES algorithm)

I'm trying to understand the AES algorithm in order to implement this (on my own) in Java code. In the algorithm all byte values will be presented as the concatenation of its individual bit values (0 ...