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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0
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0answers
17 views

How to prove that Bezier(t) polynomial lies in convex hull of points (i/n,ai) for i from 1 to n

I think i should prove firstly that: Bn,$x(t)$ for t between $0$ and $1$ lies inside the convex hull of the points $(k/n, xk)$. I know only that$ k/n$ = max between $0$ and $1$ and i found that Bezier ...
2
votes
1answer
80 views

Solution of $x^2e^x = y$

The other day, I came across the problem (or something that reduced to the problem): Solve for $x$ in terms of $y$ and $e$: $$x^2e^x=y$$ I tried for a while to solve it with logarithms, roots, and ...
0
votes
0answers
28 views

Number of roots of a multi-variate polynomial in an integral domain D. [closed]

Let D be an integral domain, and let $ g\in D[X_1,...,X_n], $ with Deg($g$)= $k\ge 0$. Let $S$ be a finite, non-empty subset of D. Show that the number of elements $\left( x_1,...,x_n\right)\in ...
0
votes
1answer
16 views

Polynomial division proof

What first? I don't know where should I begin: $a$ and $b$ is integer and $W$ is a polynomial with integer coefficients. Prove that: $a-b$ divides $W(a) - W(b)$
0
votes
1answer
27 views

Construct a degree $n$ polynomial with roots $a_1, a_2, a_3, \ldots, a_n$

We have the numbers: $a_1, a_2, a_3, \ldots, a_n$ Show that there is a polynomial $P(x)$ of degree $n$ such that $a_1, a_2, a_3, \ldots, a_n$ are roots of $P(x)$
-1
votes
1answer
33 views

A monic polynomial which does not have a root in unit disc [closed]

Let $P$ be a real polynomial of the form $P(x)=x^n+a_{n-1} x^{n-1}+\cdots+a_1 x-1$. Suppose that $P$ has no roots in the open unit disc and $P(-1)=0$. Then 1. P(1)=0 2. P(x) goes to infinity when x ...
-1
votes
1answer
34 views

Under what conditions this equation $F(x)=-c x^{n+p} - b x^n + (a-c)x^p -b =0$ has solutions?

Let $$ F(x)=-c x^{n+p} - b x^n + (a-c)x^p -b $$ where $$ (\mathscr{H})\left\{ \begin{array}{@{}ll@{}} x > 0, & \\ a > c > 0, & \\ b > 0, & \\ 0<p ...
1
vote
1answer
42 views

Constructing an irreducible polynomial of degree $2$ over $\mathbb{F}_p$

I want to construct an irreducible polynomial of degree $2$ over $\mathbb{F}_p$ where $p$ is a prime that can be written as $4k+1$. My attempt is as follow: we can assume that this polynomial is of ...
0
votes
1answer
14 views

Significance of derivative in finding square free decomposition

If $gcd(f(x),`f(x))=1$ then f(x) is square free. But what is the reason behind taking derivative of f(x)? How one came to this conclusion?
0
votes
1answer
28 views

Solving a cubic function with P and Q

I have been struggling a little bit over solving cubic functions. I have been trying to use the P and Q method. So the question is What is the approximate value of the greatest zero of $f(x) = x^3 - ...
3
votes
2answers
35 views

Number of real roots of $f ' ( x )$

Let $$f(x)=(x-a)(x-b)^3(x-c)^5(x-d)^7 $$ where $a,b,c,d$ are real numbers with $a < b < c < d$ . Thus $ f ( x )$ has $16$ real roots counting multiplicities and among them $4$ are ...
1
vote
0answers
35 views

Characteristic polynomial of matrix

If I wanted to find the eigenvalues of a matrix $\mathbf{A}$, then I could use these two options. $$\lambda\mathbf{I}-\mathbf{A}=\mathbf{0}$$ $$\mathbf{A}-\lambda\mathbf{I}=\mathbf{0}$$ However, ...
1
vote
1answer
57 views

How many roots does the polynomial $p(z) = z^8 + 3z^7 + 6z^2 + 1$ have inside the annulus $1 < |z| < 2$?

How many roots does the polynomial $p(z) = z^8 + 3z^7 + 6z^2 + 1$ have inside the annulus $1 < |z| < 2$? I know I can use Rouche's Theorem. I'm just not sure how. It states that $|f(z) − g(z)| ...
-1
votes
0answers
34 views

Primitive element in multivariate Galois field [closed]

On Singular CAS I can define a Galois field $(2^3)$ with $(x,y,z)$ variables. But I am not able to understand how $a^3+a+1$ is still its primitive element. General example taken in books is always ...
0
votes
0answers
32 views

Math Remainder Theorem Question [closed]

If $2x^n+ax^2-6$ leaves a remainder of $-7$ when divided by $x-1$ and $129$ when divided by $x+3$, find $a$ and $n$.
2
votes
1answer
36 views

Modulus of roots of polynomial tend to infinity

Define $f_n:\mathbb{C}\to\mathbb{C}$ and $(\alpha_n)$ such that:$$f_n(z)=\sum_{k=0}^n \frac{z^k}{k!}$$ and $f_n(\alpha_n)=0$. Prove $|\alpha_n|\to\infty$ as $n\to\infty$. I guess this makes sense ...
0
votes
1answer
23 views

Estimation for points in a neighbourhood of a root of a polynomial

Let $p(x)$ be a polynomial with complex coefficients and $p(\tilde x)=0$. Choose $\delta>0$ small enough, such that $\tilde x$ is the only root of $p$ in $B_\delta(\tilde x)$. I want to show that ...
4
votes
2answers
64 views

Why is a polynomial $f(x)$ sum of squares if $f(x)>0 $ for all real values of $x$?

If a polynomial $f(x)>0 $ for all real values of $x$, then $f(x)$ is sum of squares. Why is this true ? I understand that the roots of this $f(x)$ will be complex and hence will exist as ...
0
votes
1answer
46 views

Prove that the roots are equal

Suppose that all roots of the polynomial equation $x^4 - 4x^3 + ax^2 +bx + 1 = 0$ are positive real numbers. Show that all roots of the polynomial are equal. I am not getting any idea as to how to ...
0
votes
0answers
82 views

Why is the Galois group of the polynomial $f(x)=x^5-x-1$ isomorphic to $S_5$?

Currently I am trying to understand why quintic (and higher degree) polynomials are not soluble by radicals, and one of the easiest examples of a quintic polynomial which has a nonsoluble Galois ...
1
vote
1answer
21 views

Second degree polynomials in one variable (with integer coefficients) and limiting behavior of the number of prime values they take

As far as I know, we still do not have a proof that some second degree polynomial in one variable with integer coefficients takes an infinite number of prime numbers as its values, even the "simplest" ...
6
votes
3answers
84 views

Polynomial equation: $P(\sin t) = P(\cos t)$

Let $P(X)$ be a polynomial with real coefficients such that $P(\sin t) = P(\cos t), \, \forall t \in \mathbb R$. Prove that there exists a unique polynomial $Q(Y)$ with real coefficients, such that ...
5
votes
1answer
54 views

Prove a polynomial identity

Define a sequence of polynomials in the following way: $P_m(t)=\frac {1} {m!}\cdot t\cdot (t-1)\cdot...\cdot (t-m+1) $. (Where $P_0(t)=1$). I'm trying to prove the following identity: $\frac d ...
0
votes
1answer
34 views

A division problem and its linearity of reminders

If the computation of remainder by division of $x_i$ by $y_1, \dots, y_n$ is $r_i$ for $i = 1,2$. Then for every scalars $c_1,c_2$, the remainder by division of $c_1x_1 + c_2x_2$ by $y_1,\dots,y_n$ is ...
0
votes
1answer
22 views

Lower bound for third order polynomial over integers

I have the following polynomial: $P(a,b,c,d) = -2 a + 3 a b + 3 a^2 b + 6 a c + 6 b c + 2 d + 3 a d - 3 a^2 d - 3 b d - 6 a b d - 3 d^2 + 3 b d^2 + d^3 \;,$ where $a$, $b$, $c$, $d$ are integers ...
2
votes
0answers
28 views

A question on polynomials.

Let a polynomial $f\in\mathbb{R}[x,y]$, and $f(x,y)=(x^2+y^2)p(x,y)^2-q(x,y)^2$ and $p,q$ are coprime to each other. When do, $f$ and $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial ...
0
votes
1answer
36 views

Irreducible question

Can anyone show me how to prove that $y-x^3$ is irreducible in $\mathbb{A}^2(\mathbb{C})$ For my clarity, the questions asks to decompose variety $V(xy^4-x^7y^2) \subset \mathbb{A}^2(\mathbb{C}) $ ...
1
vote
0answers
24 views

Hermite Polynomial

In a famous paper by Ait-Sahalia I have found this expression for the Hermite polynomial (pp 252, line -5): $$ H_{j+1}^{\prime}\left(z\right)=-(1+j)\,H_j(z)\quad (1) $$ where $H_j$ is the $j$-th ...
0
votes
0answers
9 views

When does a system of n symmetric polynomials in n variables have exactly one solution over C up to permutation?

I was slightly amused that if I never learned about polynomials and was asked if Vieta's system of equations has exactly one solution up to permutation, the solution would be to develop polynomials in ...
3
votes
2answers
81 views

Solve the equation $x^3-6x-6=0$

Evaluate the roots of $$x^3-6x-6=0$$ I solved it using Cardano's method, but I'm looking for other elementary approaches through substitutions and properties of polynomials. ...
3
votes
2answers
47 views

Let $(a_n)_{n \geq 0}$ be a strictly decreasing sequence of positive real numbers , and let $z \in \mathbb C$ , $|z| < 1$.

Let $(a_n)_{n \geq 0}$ be a strictly decreasing sequence of positive real numbers , and let $z \in \mathbb C$ , $|z| < 1$. Prove that the sum $a_0 + a_1z + a_2z^2 + \cdots + a_nz^n +\cdots $ is ...
0
votes
1answer
14 views

Counting monomials with $k$ variables

Say we expand $\left(\sum_{i=1}^n x_i\right)^k$ into monomials. If $k=3$ there are $3n(n-1)$ monomials with two variables: $3x_1x_2^2 + 3x_1x_3^2 +\dots + 3x_1^2x_2 + \dots$. Is there a closed form ...
9
votes
2answers
100 views

Fundamental Theorem of Algebra for highschool

My teacher has told me about the Fundamental Theorem of Algebra, but I can't seem to find any proofs on it which I can understand. For something so important I'm hoping to find a proof that a ...
1
vote
3answers
96 views

How to solve $x^3 = 1$?

My intuitive side tells me to take the cube root of both the sides and get the answer $1$. However, I realize that it might be a problem for I'll lose solutions as given here: Is it the case that ...
4
votes
2answers
82 views

Express $1/(x-1)$ in the form $ax^2+bx+c$

Let $x$ be a root of $f=t^3-t^2+t+2 \in \mathbb{Q}[t]$ and $K=\mathbb{Q}(x)$. Express $\frac{1}{x-1}$ in the form $ax^2+bx+c$, where $a,b,c\in \mathbb{Q}$. I have proved that $f$ is the minimal ...
2
votes
1answer
53 views

Asymptotic behavior of integrals of Legendre polynomials

By definition $\int_{-1}^1 |P_n(x)|^2 dx = O(n^{-1})$. What about the other powers? Do we know how $\int_{-1}^1 |P_n(x)|^k dx$ behaves for any $k$? Maybe $O(n^{-k/2})$?
-1
votes
0answers
18 views

maximization by vector

I would like to maximize the below equation $C_i = log_2 (1+ \frac{1}{x} h_i g_i g_i^H h_i^H )$ h and g both of them are vectors. constraints $\sum_{i=1}^N g_i ≤ N , N=1$ i am trying to to find ...
0
votes
0answers
14 views

Change of variable in biharmonic equation

I'm currently studying how to derive Michelle's Solution for plane elasticity in the cylindrical coordinate system. I have stumbled for days to understand how the following equation: ...
0
votes
2answers
33 views

Brute force roots of a univariate polynomial

I am given a polynomial, where (a, b, c) are integers (positive and negative). $$ ax^2+bx+c $$ I need to create a simple brute force method to find the roots of this univariate polynomial. Is ...
3
votes
2answers
27 views

Find the cubic polynomial given linear reminders after division by quadratic polynomials?

A cubic polynomial gives remainders $(13x-2)$ and $(-1-7x)$ when divide by $x^2-x-3$ and $x^2-2x+5$ respectively. Find the polynomial. I have written this as: $P(x)=(x^2-x-3)Q(x)+(13x-2)$ ...
0
votes
1answer
16 views

What is the theory of finding roots of a polynomial equation by looking at the factors of the $a_n$ and $a_0$ term called?

This is commonly taught in high schools in the context of factoring polynomials. I remember this method even has its own wikipedia page (with a proof) but I forget what was the theory called. Could ...
0
votes
1answer
35 views

Show that some monomial ideal is primary

Show that $I=(X_{k_1}^{a_1},...,X_{k_s}^{a_s})$ is $(X_{k_1},...,X_{k_s})$-primary. I noticed that ...
0
votes
0answers
12 views

Upper bound on the remainder of a polynomial (not taylor)

There are many ways of approximating a function with a polynomial, $\widehat{f}(x)\approx f(x)$. One way is the taylor polynomial. A nice property that goes along with the taylor polynomial is an ...
1
vote
0answers
15 views

A problem with polynomials involving Mahler's measure

Let $f(x) = a_dx^d + \dots + a_0 = a_d(x-\alpha_1)\cdots(x-\alpha_d)$ be a polynomial with complex coefficients and (possibly multiple) roots $\alpha_i$. Define the Mahler measure $M(f)$ and length ...
11
votes
4answers
1k views

How do you find the turning points of a polynomial without using calculus?

I have a polynomial $P(x) = -x^3+12x+3$, and I am asked to find the turning points of it, and hence state how many zeroes it has. Since this chapter is separate from calculus, we are expected to solve ...
1
vote
1answer
20 views

If $f(c_i)=g(c_i)$ for $i=0,1,…,n$, prove that $f(x)=g(x)$ in $F[x]$.

Here is a problem I'm trying to solve: Let $F$ be a field. Let $f(x),g(x)\in F[x]$ have degree $\leq n$ and let each $c_i$ be a distinct element of $F$. If $f(c_i)=g(c_i)$ for $i=0,1,...,n$, ...
-1
votes
2answers
47 views

How to perform long division on polynomials? [closed]

$$\frac{x^6 - 3x^5 + x^4 - 2x^3 - 3x^2 + x - 3}{x^2 + 1}$$ I got the answer in the book, but I can't figure out how it comes up with the x for $$\ { -2x^3 + 3x^3 }$$
1
vote
2answers
38 views

Summation over a floor function of a first degree polynomial

I've been trying to solve a difficult programming question for the last four days. I've gotten most of it done, but the piece I can't seem to figure out is this: Find a closed form expression of ...
0
votes
1answer
19 views

2nd order polynomial - finding the $x$ value of the top

I have a 2nd order polynome ($y = ax^2+bx+c$) from with I know $2.5$ points. $(450,5), (600,40)$ and $(q,0)$. I know also that $(q,0)$ is the top and $q<450$. How do I solve $q$?
1
vote
0answers
38 views

Basis of the space of homogeneous polynomials clarification

I want to prove the following proposition. Let $H(n,m)$ denote the vector space of homogeneous polynomials of degree $m$ in $n$ variables over $\mathbb{C}$. Then here exist a finite number of ...