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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1answer
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Find the coordinate matrix of a polynomial with respect to a non-standard basis

I'm stuck on this question here: Find the coordinate matrix of $2-4x-3x^2$ with respect to $B = {2, x^2-1, 1-2x-x^2}$ I did the following: $a(2) + b(x^2 - 1) + c(1-2x-x^2) = 2-4x-3x^2$ But now I'm ...
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3answers
110 views

Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots.

Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots. My attempt: Consider the polynomial $g=(x-1)(1+x+x^2+x^3+\cdots+x^n)$ As $f\mid g, g$ all the roots of $f$ are roots of $g$. This means I ...
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1answer
54 views

Working in $\mathbb Q[x]$. Two polynomials are coprime if their gcd is a constant?

When are two polynomials coprime? Is it when their gcd is a constant? If we divide $x^3-7x-5$ by $x-4$, we get: $$x^3-7x-5=(x-4)(x^2+4x+9)+31$$ So, is $31$ their gcd, but since $31$ is not monic ...
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3answers
82 views

What are the values of $p$ so that equation $x^3+(p-2)x^2+(5-2p)x-10=0$ has exactly $2$ real roots…

I found this question in a maths-group in Facebook- What are the values of $p$ so that equation $x^3+(p-2)x^2+(5-2p)x-10=0$ has exactly $2$ real roots........ I think we do not count repeated roots ...
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3answers
40 views

Factor this polynomial into linear factors with coefficients in $F = \mathbb{Q}(2^{1/3}, i\sqrt{3})$

The polynomial is this: $x^3 -2$ Okay, so first I can create my field extension. I can easily extend the field to $2^{1/3}$. And I know the elements of the extension of $\mathbb{Q}(2^{1/3})$ can be ...
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2answers
430 views

Application of Taylor's Theorem in Number Theory

I'm working through Alan Baker's book A Concise Introduction to the Theory of Numbers, and there's an assertion in there that confuses me. Here's the quote: It is easily seen that no polynomial ...
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1answer
26 views

a question about field theory and polynomials

Hello all I was given this question in my field theory class on which I would certainly appreciate the help: I am given a field F of characteristic p ($ ch(F) > 0 $) and this polynomial $ f(x) = ...
2
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1answer
34 views

How do I derive this polygonal function from sample values?

I have 4 parameters with 16 sample data points each. When I plot them, I get this: The curves lead me to suspect that all these of 64 data point are derived from one polygonal function with 4 ...
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1answer
32 views

Let $f(x)=7x^{32}+5x^{22}+3x^{12}+x^2$. Find the remainder when $x^2+1$ divides $f(x)$ and $xf(x)$.

Let $f(x)=7x^{32}+5x^{22}+3x^{12}+x^2$. Find the remainder when $x^2+1$ divides $f(x)$ and $xf(x)$. I tried this problem two ways, substituting $x=1,-1$ in $f(x)$ to find the remainder, and by long ...
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0answers
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Show that $G$ is a Groebner bases of $I$ if division of $f$ on $G$ is zero for all $f\in I$.

Let $I=\langle g_1,g_2,\dots, g_t\rangle$ be an ideal in $k[x_1,\dots,x_n]$ with $k$ a field. Let $G=\{g_1,\dots,g_t\}$ be a bases for $I$. Show that if the remainder of $f$ on division by $G$ is $0$ ...
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1answer
14 views

Question about irreducible polynomials?

Is this polynomial: $irr(\sqrt{3 -\sqrt{6}}, \mathbb{Q})$ irreducible? Here is what I did $ a = \sqrt{3 -\sqrt{6}}$ $a^2 = 3 - \sqrt{6}$ $a^2 - 3 = -\sqrt{6}$ $(a^2 - 3)^2 = 6$ Our polynomial ...
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2answers
30 views

Suppose $f(x)$ is a polynomial of degree 5, and with leading coefficient 1. [closed]

Suppose $f(x)$ is a polynomial of degree $5$, and with leading coefficient 1. If further that $f(1)=1, \ f(2)=2, \ f(3)=3, \ f(4)=4, \ f(5)=5$. What is the value of $f(6)$?
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1answer
66 views

Number of complex roots of a degree 6 polynomial

Given some degree 6 polynomial $f(x) \in \mathbb{Q}[x]$, is there any invariant of the polynomial (depending on the coefficents) that will tell you if this polynomial has 6 complex roots or just 2 ...
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1answer
29 views

Residue class ring $\mathbb{Z}[x]$/I and $\mathbb{Z}[x]$/J

$I = \left\lbrace \sum_{i=1}^{n} a_ix^i : n \in \mathbb{N}, a_1, ..., a_n \in \mathbb{Z} \right\rbrace$ beeing an ideal of $\mathbb{Z}[x]$ with polynomials without a constant term and $J = ...
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0answers
19 views

Find eigenvalues of a matrix using Perron–Frobenius theorem

I have to find the largest eigenvalue of a matrix containing only positive entries: $$\left( \begin{array}{ccc} e^{a} & 1 & e^{-a} \\ 1 & 1 & 1 \\ e^{-a} & 1 & e^{a} ...
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1answer
24 views

What is Horner's Method of synthetic division… [closed]

What is Horner's method of synthetic division. Is there is any proof of this method??? How does it works?
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0answers
22 views

How do I evaluate this expression? Product inside sum.

Let $a_0,a_1,\ldots,a_n,b_0,b_1,\ldots,b_n \in \Bbb{C}, n\geq 1$ such that $a_i\neq a_j \forall i\neq j$. Prove that the polynomial: $$ f=\sum_{k=0}^n b_k \left(\prod_{\substack{0\leq j \leq ...
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2answers
25 views

What does $p(x)=p(1-x)$ in $P=\{p(x)\in {{R}_{4}}[x]|p(x)=p(1-x)\}$ mean? [closed]

What does $p(x)=p(1-x)$ in $P=\{p(x)\in {{R}_{4}}[x]|p(x)=p(1-x)\}$ mean?
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2answers
45 views

What exactly is the purpose of the evaluation homomorphism?

I just don't understand the point of terming the evaluation of a polynomial by a map like this? And what's more, the map is going into a larger field than the field the polynomial is in anyway. What ...
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1answer
21 views

Help finishing proof with polynomial discriminant?

Prove that the discriminant of $$f(x) = x^n + nx^{n-1} + n(n-1)x^{n-2} + \cdots + n(n-1)\ldots (3)(2)x + n!$$ is $(-1)^{n(n-1)/2}(n!)^n$. So far, I let $\alpha_1,\ldots, \alpha_n$ be the roots of ...
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2answers
96 views

How to show $f(x)$ has no root within $\Bbb Q$

A polynomial problem from my old algebra textbook: $f(x)\in\Bbb Z[x]$ with leading coefficient $1$, $\deg f(x)\ge 1$, and both $f(0)$ and $f(1)$ are odd numbers, prove: $f(x)$ has no root within ...
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0answers
32 views

Two-term asymptotic approximation for roots of a polynomial (dominant balance)

I'm trying to find the roots to the following equation: $t^5 - \epsilon t^3 + \epsilon^3 = 0$ as $\epsilon \rightarrow 0$. From expansion $t= \epsilon^{\alpha}t_1 + \epsilon^{2\alpha}t_2 + ...
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1answer
30 views

On a question about polynomial ring

Let the ring $ R$ define as the following $R=\{a_1+a_2x^2+a_3x^3+...+a_x^n;a_i\in \mathbb R,\,n\gt 2\}$ and Let the ideal $I$ generated by $<x^2+1,x^3+1>$. Is $I=R$ or not?
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2answers
58 views

Solve the equation -

Solve $$ 3-\frac{4}{9^x}-\frac{4}{81^x}=0 $$ I had this question for an exam today and I want to find out if my answer was correct.
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3answers
51 views

Let $f(x) =7x^{32}+5x^{22}+3x^{12}+x^2$. Then find its remainder in the following cases.

Let $f(x) =7x^{32}+5x^{22}+3x^{12}+x^2$. (i) Then find the remainder when $f(x)$ is divided by $[x^2+1]$. (ii) Also find the remainder when $xf(x)$ is divided by $[x^2+1]$. Given both the ...
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0answers
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Question about principle ideals and polynomials and quotient ring construction?

Say I have a ring of polynomials in $R[x]$. I wish to define the quotient group $R[x]/<x^2+1>$. My question lies in the ideal generated by $<x^2 + 1>$. This is the set of all numbers such ...
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6answers
66 views

Prove this polynomial falls within $\mathbb R[x]$

[ The problem below is from Yao Musheng (姚慕生), Wu Quanshui (吴泉水), Advanced Algebra (高等代数学) Ed $2$, Fudan University Press, page $207$. ] Suppose $f(x)\in \mathbb C[x]$. If $\forall c\in \mathbb ...
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3answers
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Finding real coefficients of equation given that $a+ib$ is a root

Below is the question present in a past examination paper. I'll be giving my attempts and how I thought it through. Do feel free to point out any mistakes I make throughout my working even if ...
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1answer
21 views

Series expansions of inverse polynomials

Suppose one is given a strictly monotonous polynomial, $$f(x) = \sum_{n=0}^N a_n x^n$$ So that for a given $y$ there exists a single real $x=f^{-1}(y)$. It would be nice* to be able to calculate the ...
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1answer
12 views

Do Bezier control points aproximate their curve?

I was just reading here about degree elevation in Bezier curves and I noticed that in the diagrams of the progressively higher degree curve, that the control points began to approximate the curve ...
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1answer
38 views

Methods for factorising polynomials into irreducibles over finite fields

I was given a problem recently, part of whose solution was to factorise $x^{15}+1$ in $\mathbb F_2[x]$. It turns out that the factorisation is $$(x+1)(x^2+x+1)(x^4+x^3+x^2+x+1)(x^4+x+1)(x^4+x^3+1),$$ ...
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3answers
70 views

Distinct roots of $z^n-z$

How would we prove that for any positive integer $n$ the complex roots of $z^n-z$ are all distinct? In the case that $n=1,2,3$ I have factored it directly but how can we do it in general?
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1answer
37 views

How can one solve an equation over over a specific finite field?

How can one solve an equation of the following form where the coefficients are in $GF(2^{128})$? $Az^3 + Bz^2 + Cz + D = 0$ The operations are defined over the same field.
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proof that $\frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1}$ is irreducible

I am reading the group theory text of Eugene Dickson. Theorem 33 shows this polynomial is irreducible $$ \frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1} \in \mathbb{Z}[x]$$ He shows this polynomial ...
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1answer
32 views

Minimal polynomials

Can someone explain to me how the minimal polynomials in page 4 of this document are obtained? Please help me. http://web.ntpu.edu.tw/~yshan/BCH_code.pdf It should be something standard about ...
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1answer
60 views

If a degree $n$ polynomial has $f(k)=k/(k+1)$ for $k=0,1,\ldots,n$, what is $f(n+1)$?

Suppose $f(x)\in K[x]$, $\deg f(x)=n$ and $f(k)=k/(k+1)$ for any $k=0,1,2,\ldots ,n$, what is $f(n+1)$? I have a vague memory that there is a very clever trick that can solve this problem easily, ...
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0answers
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Looking for polynomial to represent approximate 2D matrix.

I am looking for a polynomial that similars Legender polynomial(a set of orthogonal polynomial basis function. Could you suggest to me some polynomial? Because my goal is that I want to approximate 2D ...
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1answer
60 views

finding the remainder of $(x+1)^7+x^5+(x-1)^3$ divided by $ x+2$

How can i find the remainder of $(x+1)^7+x^5+(x-1)^3$ divided by $x+2$? I tried long division but it's really messy. Also i saw that $x=0$ is a root but it still difficult.
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1answer
43 views

Factorising polynomials over $\mathbb{Z}_2$

Is there some fast way to determine whether a polynomial divides another in $\mathbb{Z}_2$? Is there some fast way to factor polynomials in $\mathbb{Z}_2$ into irreducible polynomials? Is there a ...
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1answer
31 views

Finding the maximum of sum of coefficients of a polynomial

Suppose $p(x)=ax^2+bx+c$ is a quadratic polynomial with real coefficients and $|p(x)| \leq 1$ for all values of $x$ in the range $[0,1]$. Prove that maximum possible value of $|a|+|b|+|c|$ is $17$. ...
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0answers
16 views

How to assemble rook polynomials?

I have a problem. I need to assemble a rook polynomial for the chessboard (6x6 boards). Black boards are 1, white boards are 0. ...
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2answers
74 views

$x^9 - 2x^7 + 1 > 0$

$x^9 - 2x^7 + 1 > 0$ Solve in real numbers. How would I do this without a graphing calculator or any graphing application? I only see a $(x-1)$ root and nothing else, can't really factor an ...
3
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1answer
89 views

Reducible polynomials in $\mathbb{Z}[X]$

Let $(a_n)_{n\geq 1}$ be a strictly increasing sequence of integers and $k$ a non-zero integer such that for some $N\in \Bbb Z^+$, the polynomial $$ p_{N}(x)=(X-a_1)(X-a_2)\cdots(X-a_N)+k $$ ...
2
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1answer
36 views

Total derivative for a polynomial

I refer to Rudin's (Principles of Mathematical analysis, 3rd ed.) definition of differentiability: Suppose E is an open set in $R^n$ and f maps E into $R^m$ and $x \in E$. If there exists a linear ...
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4answers
330 views

Given $f(1)=10,f(2)=20,f(3)=30$ find $f(12)+f(-8)$ for a 4-th degree monic polynomial

If $f(x)=x^4+ax^3+bx^2+cx+d$. Given $f(1)=10,f(2)=20,f(3)=30$ find $f(12)+f(-8)$. This problem has troubled me a lot.The more I try to solve it,it becomes lengthier. My problem is that there are four ...
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1answer
44 views

Laguerre theorem

I'm looking for a proof of the theorem 7, page 6, of this document : http://www.nipne.ro/rjp/2013_58_9-10/1428_1435.pdf Theorem 7 (E. Laguerre) Let $f \in \mathbb{R}[x]$ be a polynomial of degree ...
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1answer
55 views

Simplifying a sum of products related to Vandermonde determinant

How to show this equality? $$ 1=(-1)^n\sum_{k=0}^n\frac{x_k^n}{\prod_{\substack{l=0 \\ l \neq k}}^n(x_l-x_k)} $$ This is part of a proof to show the value of the determinant of the Vandermonde matrix ...
2
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2answers
85 views

Is $x^4+nx+1$ irreducible?

Consider the polynomial $\xi= x^4+nx+1\in \mathbb Z[x]$. Show that if $n=\pm2$ then $\xi$ is reducible and that $n\neq\pm2$ implies $\xi$ is irreducible. I got the answer by writing the ...
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1answer
54 views

Polynomial satisfying $f(x) = f'(x) \cdot f''(x)$

If a polynomial of degree $n$ satisfies $f(x) = f'(x)\cdot f''(x)$ (such that $n$ belongs to $\mathbb R$) then $f(x)$ is? A) an onto function B) an into function C) no such function possible D) ...