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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2
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2answers
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Showing $\sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}} = \frac{\sqrt{13}}{64}$

I would like to show that $$ \sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}} = \frac{\sqrt{13}}{64} $$ I've been working on this for a few ...
0
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2answers
21 views

Polynomial identity for a sum

If $$f(x) = \sum_{i=0}^{n}A_i x^i \quad \text{ and } \quad g(x) = \sum_{i=0}^{n}B_i x^i$$ are two degree $n$ polynomials, then we can say that the polynomial $$h(x) = \sum_{k=0}^{2n}C_k x^k \quad ...
0
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1answer
7 views

Find the parameter $\alpha$ that …

My question is: For which value of the real parameter $\alpha$ the following equation has a root with the multiplicity higher than $1$. $$3x^4+4x^3-6x^2-12x+\alpha=0$$ $Thank $ $you$ $!!!$
2
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2answers
29 views

Minimal polynomial over Q

Let $\omega$ be a primitive 7th root of 1 over $\Bbb Q$ .Let $\alpha= \omega+\omega^6$. Find the minimum polynomial of $\alpha$ over $\Bbb Q$. What I have so far is; $\omega^7=1$ ...
0
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0answers
15 views

Invertibility of a Polynomial map.

Given following polynomial map $f:\mathbb{R}^2\to V\subset \mathbb{R}^3 $: $$ (z_1,z_2)\mapsto (2z_1-z_2, 2z_1^2-z_2^2, 2z_1^3-z_2^3) $$ Is this map a bijection? If so, how?
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0answers
19 views

A polynomial identity

Let $x_1<x_2<\dots<x_n$ be $n$ real numbers. I'm trying to prove the following polynomial identity : $$ P(Y):=1+Y+Y^2+\cdots+Y^{n-1}= \sum_{k=1}^n \prod_{\underset{j\neq k}{1\le j \le n}} ...
0
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1answer
13 views

Is there a way to compute if(i < j) k := (a + b)c with polynomials over $\Bbb{Z}_p$?

Let $p$ be a prime and let all variables be in $\Bbb{Z}_p$. Then you can write the result of if(i > 0) k = (a + b)c; (C code) as a polynomial $k := ...
2
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4answers
41 views

Limit of a rational function to the power of x

Ok so I have been trying for days already to find a solution to this all around the web and in math books but to no success. The problem is to evaluate a limit of a function composed by polynomial ...
0
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4answers
78 views

Why all such polynomials have $-1$ as a root?

Why all polynomials of this form have $-1$ as a root? $ x^5+x^4+x^3+x^2+x+1 $ and similar polynomials like $ x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$
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0answers
14 views

PSD matrix and non-negative polynomial

So I'm trying to prove that if there exists a $5 \times 5$ matrix $Q$ such that $$Q \succeq0,\,\, a_{l-1} = \sum\limits_{i+j=l} Q_{ij} , l=1,\ldots,5$$ then there exists a fourth degree polynomial ...
0
votes
1answer
11 views

How do I distribute this polynomial expansion?

Ok, so for some reason, I cannot seem to get this simple polynomial multiplication correct no matter how many times I do it. I am working in $\mathbb{Z}/13\mathbb{Z}$. $$ (4x+11)(5x+(3x^2+1)) $$ ...
2
votes
0answers
32 views

Roots of polynomial in $F_3[x]$

Let $\alpha$ be a root of $x^2 + x + 2 = 0$ in $F_3[x]$. I am asked to show that $x^3 + x + 1$ has roots $\alpha$, $\alpha^2$ and $\alpha^4$. I started by observing that $\alpha^2 + \alpha + 2 = 0 ...
0
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0answers
9 views

Are these computational models equivalent?

Let $f : X \to Y$ be a problem that you want to compute. Say we have an $O(1)$-computable maps, $\phi, \psi$, such that $X \xrightarrow{\phi} (\Bbb{Z}_n)^k \xrightarrow{\psi} Y$. After all, ...
0
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0answers
30 views

Number of roots of a polynomial (Proof)

What might be a simple proof to show that the maximum number of roots of a polynomial is equal to the degree of the polynomial? For example a quadratic polynomial can have a maximum of 2 roots. Can ...
2
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2answers
28 views

Uniform convergence of Lagrange polynomials

There is a well-known theorem that states that on a closed interval $[a,b]$ any continuous function is the limit of a uniformly convergent sequence of polynomials. Proofs for this theorem usually ...
5
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2answers
119 views

How find this system $a^2+b^2=3,a^2+c^2+ac=4,b^2+c^2+\sqrt{3}bc=7$

Find the this system real solution $$\begin{cases} a^2+b^2=3\\ a^2+c^2+ac=4\\ b^2+c^2+\sqrt{3}bc=7 \end{cases}$$ I think that one can use Geometry to solve this system. Maybe there exist an ...
1
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1answer
44 views

Number of integer roots possible of the following polynomial

Let $p(x)$ be polynomial with integer coefficients, such that $p(0)$ and $p(1)$ are both odd. What is the maximum possible number of integer roots this polynomial can have?
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4answers
463 views

Matching of polynomial coefficients

I am trying to find the proof/theorem that states: Given two polynomials in x, if they are equal to eachother, their coefficients must also be equal For example, in ax^3 + bx^2 + cx + d = ...
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1answer
26 views

How can i optimize this type of equation?

Given an equation, a polynomial for example, how can i optimize it? see the equation below. $$y = -0.266x^6 + 48.19x^5 - 3424.x^4 + 12170x^3 - (2\times 10^6)x^2 + (2\times 10^7)x - (6\times 10^7)$$ ...
3
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1answer
401 views

Characteristic Polynomial of a Linear Map

I am hoping for some help with this question from a practice exam I am doing before a linear algebra final. Let $T_1, T_2$ be the linear maps from $C^{\infty}(\mathbb{R})$ to ...
0
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3answers
33 views

Find all polynomials p with real coefficients

Find all polynomials $p$ with real coefficients such that $p(x+1)=p(x)+2x+1$. I feel like in this question you let $x+1=x'$.
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2answers
32 views

Addition in $\operatorname{GF}(2^4)$

How can I compute $A(x)+B(x) \mod P(x)$ in $\operatorname{GF}(2^4)$ using the irreducible polynomial $P(x)=x^4+x+1$. What is the influence of the choice of the reduction polynomial on the computation? ...
6
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1answer
79 views

Show that $P$ is divided to simple roots knowing that $a_{k}^2-4a_{k-1}a_{k+1}>0$

Let $P(X)=a_0+a_1X+..+a_nX^n\in R[X]$ Assume that $\forall k, a_k>0$ and $a_{k}^2-4a_{k-1}a_{k+1}>0$ Show that $P$ is divided to simple roots in $R[X]$. i.e. ...
0
votes
3answers
24 views

Coefficients of even powers

Just a simple thought experiment I was running in my head. Say I have a nonnegative even degree polynomial such as $f(x) = ax^4 + bx^3 + cx^2 + dx + e$. Is it true that the coefficients of the even ...
0
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2answers
34 views

finding the equation of a polynomial given its graph

I have a graph of polynomial and I would like to know how to determine its equation. Please, this isn't homework. What I'd like to do is actually reproduce this graph. Thanks.
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1answer
10 views

Splitting a multivariable polynomial into homogeneous components

In Wikipedia's proof of the fundamental theorem of symmetric polynomials, it states that the proof focuses on the case where the polynomial is homogeneous, and that "The general case then follows by ...
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0answers
24 views

Divisibility by $z-z_0$ if $z_0\in \mathbb{C}$ [duplicate]

I have a problem I'm working on, and I'm just not getting it. Suppose that $z_0\in\mathbb{C}$ is fixed. Show that if $P(z)=c(z^k-z_0^k)$, then there exists a polynomial $Q(z)$ such that ...
0
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2answers
18 views

Non negativity condition for quartic polynomials?

Say I have a quartic polynomial $f(x) = ax^4 + bx^3 + cx^2 + d$. I am told that $f(x)$ is nonnegative iff it can be expressed as a sum of squares as follows. $f(x) = \sum_{i=1}^4 q_i(x)^2$. As an ...
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2answers
180 views
+100

Help with Proof explanation

I need help in understanding a (topological) proof of Fundamental theorem of algebra. Here is the Proof: Suppose $f(z)=a_nz^n+...+a_0$ with $a_0 \neq 0, n \geq1.$ WLOG, assume that $a_n=1.$ We ...
96
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14answers
8k views

Why can ALL quadratic equations be solved by the quadratic formula?

In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
1
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2answers
33 views

Polynomials division algebra problem

Find sum of coefficients of the quotient obtained in: $$\frac{2x^n+x^{n-1}+x^{n-2}+...+x^2+x+5}{x-\frac{1}{2}}$$ I got "n" as the answer but according to the book is wrong, I don't know what is ...
0
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0answers
36 views

Existence of a det. poly-time algo for problem $f: X \to Y$.

$f : X \to Y$ is a deterministic polynomial-time algorithm for problem inputs $x \in X$ and problem outputs $f(x) = y \in Y \iff $there exists a polynomial $P_f \in \Bbb{Z}[x_1]$ such that $C\cdot ...
7
votes
3answers
404 views

Degree of a polynomial

If I have a polynomial, for example, $$ x^8 + x^2 + \dfrac{1}{x} $$ would this be considered to be of degree 8? I am working on a question involving the function $ \frac{1}{x} $ and I am wondering how ...
3
votes
0answers
22 views

Möbius transformation that permutes roots of a cubic polynomial

The roots of the polynomial $x^3-3x-1$ can be permuted by the function $z\mapsto \dfrac{-1}{1+z}$ which is easily checked by a direct calculation. Is there a simple formula for a Möbius ...
3
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4answers
34 views

Finding the Remainder

Given the polynomials $$P(x) = nx^n+(n-1)x^{n-1}+(n-2)x^{n-2}+\cdots+x+1$$ and $$Q(x)=x(x-1)^2$$ find the remainder of the division $\dfrac{P (x)}{Q (x)}$.
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2answers
31 views

Finding real, distinct eigenvalues for arbitrary constants

Let $A= \begin{bmatrix} 1 & 1 & 0 \\ -4 & -3 & 1 \\ k & 0 & 0 \end{bmatrix}$. Find all values of $k$ such that $A$ has three real distinct eigenvalues. I have obtained the ...
3
votes
1answer
52 views

Factors of integers of the form $k^2-k+1$

Factorisation of arbitrary integers is of course a computationally hard problem. But what if the integers I'm interested in factorising are all of the form $k^2-k+1$ ? Is there some way to compute ...
1
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0answers
65 views

Small generating set of third degree polynomials in $R=\mathbb{Z}_2[x_1,\dots,x_n]/\langle x_1^2-x_1,\dots,x_n^2-x_n\rangle$

Let $R=\mathbb{Z}_2[x_1,\dots,x_n]/\langle x_1^2-x_1,\dots,x_n^2-x_n\rangle$, i.e., we can think of $R$ as the ring of multivariate polynomials with the additional property that one can "linearize" ...
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0answers
76 views

Too many independent cubic polynomials in an ideal $I\subset \mathbb C[x,y,z]$

Let us consider the ideal $I=(x^2-x,y,xz)\subset \mathbb C[x,y,z]$. I want to prove that $I$ contains (exactly) $5$ linearly independent polynomials of degree $3$. In three variables, we have ...
6
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1answer
194 views

Why is the polynomial $S(\vec{x})$ with coefficients obeying a constraint homogeneous?

I have recently been working on a problem to prove that a particular polynomial is in fact homogeneous. Although I have found out that this is true, I am curious to see whether there might be a deeper ...
12
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1answer
211 views

Semialgebraic conditions that convey properties of Galois group

Let $f \in \mathbb{Z}[x]$ be a polynomial of degree $n$ with integer coefficients and let $G_f$ be the Galois group of $f$ over $\mathbb{Q}$. I am trying to collect results that convey some ...
2
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0answers
32 views

Linear Independence of Powers of “roots vector” [duplicate]

Let us be working over the field of complex numbers. Suppose $f(x)= a_n x^n + \cdots +a_1 x + a_0$ is a degree $n$ polynomial with $n$ distinct roots $z_1,\ldots,z_n$. Is the following matrix always ...
2
votes
1answer
38 views

Show that $(x-a,x-b)=1$

Knowing that $K$ is a field, $a,b \in K$ different from each other,show that $x-a,x-b$ co-primes. We suppose that $\exists f(x) \in K(x)$ such that: $f(x)|x-a$ and $f(x)|x-b$ Then $\deg f(x) \leq ...
3
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2answers
137 views

Nice exercises on resultants

I would like to ask if some one knows a source (a book, or lecture notes ect) that contains several nice exercises on resultants of polynomials (it would be nice if there were some solutions as well ...
2
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0answers
46 views

Consider the quadratic equation $ax^2-bx+c=0, a,b,c \in N. $ If the given equation has two distinct real root…

Problem : Consider the quadratic equation $ax^2-bx+c=0, a,b,c \in N. $ If the given equation has two distinct real roots belonging to the interval $(1,2) $ then the minimum possible values of a is ...
2
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1answer
28 views

Space of complex poynomials

Let $\mathbb{C}_n[z]$ be the space of polynomials (of degree $\le n$) with complex coefficients, let the inner product be $(p,q):=\int_{-1}^1p(t)\overline{q(t)}dt$. There is one and only one $K_{w} ...
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vote
3answers
76 views

Solving a nonlinear system of equations.

Given that $x,y,z\in\mathbb R$, solve $$\begin{cases}6x^2-12x=y^3\\6y^2-12y=z^3\\6z^2-12z=x^3\end{cases}$$ I've tried adding the equalities but to no avail. I'd add what I've tried, but it'd be ...
5
votes
1answer
75 views

Prove that the equation $1+x+\dfrac{x^2}{2!}+\cdots+\dfrac{x^n}{n!}$ cannot have a multiple root.

Prove that the equation $$1+x+\dfrac{x^2}{2!}+\cdots+\dfrac{x^n}{n!}$$cannot have a multiple root. Using induction and the result that $f(x)=0$ have a root $\alpha$ of multiplicity $r\implies ...
0
votes
1answer
43 views

dimension of subspace - polynominals evaluated on f

I need to prove that the dimension of the subspace of endomorphisms is less or equal m, if m is the degree of a polynomial p of K[t] \ {0} with p(f) = 0 (f is endomorphism). In a second step I ...
0
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1answer
17 views

A Polynomial with square values

Can I find the number of values ​​of the variable X for which the value of the polynomial $100X^2+160X+M$ is a perfect square, depending on M