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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2answers
30 views

Factorisation of a polynomial

How to factorize $a^6+8a^3+27$, I got different answers but one of the answers is $(a^2-a+3)(a^4+a^3-2a^2+3a+9)$, can someone tell me how to get this answer? Thanks
1
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0answers
31 views

Finding a linear system to solve quadratic equations

considering an equality with a polynomial of second degree where the coefficient for $x^2$ is $1$ I know that $$ a x^2 + b x + c = a(x-\alpha)(x-\beta) = 0 $$ I also know that $$ \alpha + \beta = -...
2
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1answer
27 views

Is it possible to factor out monomials in a rational function?

after thinking about the problem for some hours I thought I come here to ask. My problem is that I want to do a coordinate transformation on the following equation $y=\frac{a}{x^2}+\frac{b}{x}+c+dx+...
0
votes
3answers
42 views

Roots of polynomial with positive coefficients

My question is very simple. Suppose we have a polynomial defined as follows: $$p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots+a_0 $$ where all of the $a_n$'s are all real and positive. Is there something ...
1
vote
1answer
50 views

The graph of $f(x)=ax^2+bx+c$ contains the points $(m,0)$ and $(n,2016^2)$. How many values of $n-m$ are possible?

Let $a,b,c,m$ and $n$ be integers such that $m<n$ an define a quadratic function as $f(x)=ax^2+bx+c$ where $x$ is real. The $f(x)$ has a graph that contains the points $(m,0)$ and $(n,2016^...
2
votes
0answers
41 views

Part (a) of Exercise 3.4 of Eisenbud's Commutative Algebra

In the part (a) of Exercise 3.4, it suggests that we may use the relation: $${\text{Content}(fg)}\subset{\text{Content}(f)\text{Content}(g)}\subset{\text{rad}\left(\text{Content}(fg)\right)}$$ to ...
2
votes
1answer
88 views

Convert $x \not\equiv 0$ mod $pq$ to a modulo polynomial

$p,q \in \mathbb{P}$, primes For $x \not\equiv 0 \bmod p$ you can write $(x-1)(x-2) \dots (x - (p-1)) \equiv 0$ mod $p$ Is there a way to do the same for a a composite modulus $pq$? Note: $(x-1)(x-...
2
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0answers
15 views

sampling requirements in probabilistic polynomial identity testing

In the Schwartz–Zippel algorithm for bounded error probabilistic polynomial identity testing, the main theorem is the following: For a non-zero multivariate polynomial $p(x_1,...,x_n)$ of total ...
9
votes
3answers
123 views

Inverse of the Pascal Matrix

Let $P_n$ be the $(n+1) \times (n+1)$ matrix that contains the numbers of Pascal's triangle in the upper triangle. For example in the case of $n=3$ $$ P_3 = \begin{pmatrix} 1 & 1 & 1 & 1 \...
-1
votes
5answers
184 views

Polynomial function [on hold]

When a polynomial $f(x)$ is divided by $x-5$ or $x-3$ or $x-2$ it leaves a remainder of $1$. Which of the following would be the polynomial? a. $ x^3 - 10x^2 + 31x + 31$ b. $x^3 - 10x^2 + 31x - 31 ...
3
votes
1answer
88 views

Proof for the arithmetic progression

So I was going through a few olympiad questions, and here is a question I found Now, I found the three terms of the progression in terms of a and b, and arrived at $a^2$+ 2 b + 1 = 0. However, I'...
1
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1answer
19 views

Example of 2 trinomial multiplication which is equal to sum of 2 monomials

How to find out $P$ as an algebraic monomial which $P=ma$ and $(a^2 P+1)(a^2 P+1)$ answer be sum of two monomials $Q,R$ eg $(a^2+a+1)(a^2-a+1) = a^4 + a^2 + 1$ which is sum of three monomials. *...
0
votes
2answers
36 views

If a and b are odd integers then find the number of integral roots of $(x^{10} +ax^9 +b=0)$

If a and b are odd integers then find the number of integral roots of $(x^{10} +ax^9 +b=0)$ I've no idea how to solve this question. Any help would be appreciated. :|
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0answers
19 views

One equation that fits other/multiple equations

I have three equations, one linear, one powered, and one a 2nd order polynomial. Say these equations are: $0.5065x^{2.5066}$, $-11.185x^2+2325.1x-83917$, $729x-28736$ Edit: These are functions, ...
3
votes
2answers
38 views

Are polynomial fractions and their reductions really equal? [duplicate]

I'm reading Larson's AP Calculus textbook and in the section on limits (1.3) it suggests finding functions that "agree at all but one point" in order to evaluate limits analytically. For example, ...
1
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0answers
22 views

Valuation of discriminant

So the discriminant of a polynomial of degree $n$ in the form of determinant of the resultant matrix can be written as $$\det(D)\det(A-BD^{-1}C)$$ where $A, B, C, D$ are block matrices of the ...
8
votes
5answers
126 views

Proving that the roots of $1/(x + a_1) + 1/(x+a_2) + … + 1/(x+a_n) = 1/x$ are all real

Prove that the roots of the equation: $$\frac1{x + a_1} + \frac1{x+a_2} + \cdots + \frac1{x+a_n} = \frac1x$$ are all real, where $a_1, a_2, \ldots, a_n$ are all negative real numbers.
6
votes
2answers
2k views

Why can we use the division algorithm for $x-a$?

In Theorem 5.2.3 in these notes, it is said that Since $x − a$ has leading coefficient $1$, which is a unit, we may use the Division Algorithm... Why is this true? I thought that the Division ...
2
votes
2answers
498 views

Symbolic polynomial interpolation

I'm trying to create polynomials from some symbolic points to discretize derivations. This means I'm having data like $(a, \phi(a)),\ (b, \phi(b) ) $and $(c, \phi(c))$ and want to fit a second order ...
3
votes
2answers
508 views

Proof that the following function is a polynomial

I've been trying to get my head around this problem for a long time, yet I have not been able to make much progress. Let $\ell_0(j) = \left\lfloor \frac{1}{2}\left( \sqrt{8j^2 - 8j + 1} + 2j - 1 \...
1
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2answers
58 views

Construction of a 8-degree polynomial with 16 real numbers

(Vietnam TST 2016/6) Given $16$ distinct real numbers $\alpha_1,\alpha_2,\ldots,\alpha_{16}$. For each polynomial $P$, denote $$V(P)=P(\alpha_1)+P(\alpha_2)+\cdots+P(\alpha_{16}). $$Prove that there ...
4
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1answer
44 views

Two questions on the Gaussian integers [duplicate]

I have two questions on the Gaussian integers. Is any element in $\mathbb{Z}[i]$ the root of a monic polynomial with coefficients in $\mathbb{Z}$? Conversely, does any element in $\mathbb{Q}(i)$ ...
0
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1answer
34 views

Polynomials in $\mathbb{F}_{q}[x]$ invariant under translation of $x$

Let $p$ be prime, $r \in \mathbb{N}_{>0}$ , $q = p^r $ and $ K := \mathbb{F}_{q}$ the finite field with $q$ elements. Let $F$ be the set of polynomials, which do not change under translation: $$ ...
1
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0answers
35 views

Are constants a special case of coefficients?

What I hope to understand better, is the relation between constants and coefficients. Consider the following polynomial: $$3x^2+2x+5$$ What are the coefficients in the expression? Obviously, 3 and 2 ...
1
vote
1answer
27 views

Smooth functions can not be non-polynomial at one point

I am studying the classic problem here about Baire's Category theorem. One of the remarks is that if $f$ is smooth and not a polynomial, $$X = \big\{x : \forall(a,b)\ni x, f|_{(a,b)} \text{ is not a ...
0
votes
2answers
30 views

On the maximal of polynomial at a point

I faced this problem when I studied polynomial. Let $p(x)=ax^3+bx^2+cx+d$ be a cubic polynomial with real coefficients, and $p(5)+p(25)=1906$. Find the maximal value of $|p(15)|$. I ...
0
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0answers
11 views

relationship between independence of multivariate polynomials, generating sets of polynomial ideals

I am studying something that touches on Groebner algorithms at the moment and It seems like i am missing something obvious about the relationship between three definitions that feel like they should ...
2
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3answers
59 views

How to prove that this quartic equation has exactly 2 real roots?

So I have this quartic equation here: $x^4-3x+1=0$ I'm supposed to prove this equation has exactly 2 roots. I defined $f(x)=x^4-3x+1=0$ Then I used the Intermediate value theorem at $f(0)$ and $f(...
0
votes
1answer
50 views

Does there exist any non-zero polynomial $f:\mathbb C \to \mathbb C$ such that $f(x+2)-2f(x+1)=f(x) , \forall x \in \mathbb C$ ?

Does there exist any non-zero polynomial $f:\mathbb C \to \mathbb C$ such that $f(x+2)-2f(x+1)=f(x) , \forall x \in \mathbb C$ ?
0
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1answer
35 views

$A$ is a square complex matrix. $A^k-A=0$ for some $k\geq 2$. Prove that $A$ is diagonalizable over $\mathbb C$

$p(x)=x^k-x=x(x^{k-1}-1)$ What I want to do is to say that $(x^{k-1}-1)=(x-z_1)(x-z_2)...(x-z_{k-1})$ and therefore A is diagonalizable (because of the distinct roots in the polynomial), but i'm not ...
1
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3answers
49 views

Why the leading coefficient is positive?

Help is needed in explaining the following (partial) proof:- Let $Q(x) = ax^4 + bx^3 + cx^2 + dx + e$. Suppose “that Q(x) = 0 has no real roots. Thus, Q(x) is always positive or negative for all ...
4
votes
1answer
100 views

When is a given polynomial a square of another polynomial?

I meet a problem in which I hope to show a special polynomial is not a square of another polynomial. More precisely, let's consider the polynomial $$f(x):= 1-x+2bx^n-2bx^{n+1}-b^2x^{2n-1}+2b^2x^{2n}-b^...
3
votes
1answer
86 views

Finding the roots of an octic

I'm trying to solve a problem, but it involves finding the exact roots of the octic polynomial $$x^8+4x^7-10x^6-54x^5+9x^4+226x^3+125x^2-301x-269$$ How can I find the roots of an octic? Wolfram ...
7
votes
0answers
57 views

If $A[X] \cong B[X]$ as rings, are the degrees of irreducible polynomials the same in $A$ and in $B$?

First, I ask my question and then I add some explanations: Suppose that $A$ and $B$ are two commutative rings such that $A[X] \cong B[X]$ as rings. Denote by $D_A$ the set of all positive integers ...
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0answers
25 views

Is there a “concatenation operator” for polynomials?

Wikipedia says that the concatenation operator $\|$ concatenates digits of two numbers: ... the concatenation of 69 and 420 is 69420. Is there a similar concatenation operator (or the same?) for ...
5
votes
1answer
45 views

Does there always exist an irreducible polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$? [duplicate]

Let $p$ be a prime and let $d$ be a positive integer. Does there always exist an irreducible (i.e. unfactorable) polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$?
30
votes
2answers
8k views

Number of monic irreducible polynomials of prime degree $p$ over finite fields

Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ do exist over $F$? Thanks!
2
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3answers
107 views

For what powers $k$ is the polynomial $n^k-1$ divisible by $(n-1)^2$? [closed]

How do you prove this? $$\left(n-1\right)^2\mid\left(n^k-1\right)\Longleftrightarrow\left(n-1\right)\mid k$$
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4answers
79 views

The cubic equation $x^3-5x^2+6x-3 = 0$ has solutions $\alpha$, $\beta$ and $\gamma$. [on hold]

The cubic equation $x^3-5x^2+6x-3 = 0$ has solutions $\alpha$, $\beta$ and $\gamma$. Find the value of $$\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}$$
0
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1answer
44 views

clarify doubts about polynomial

In my math algebra class my teacher says if $$(1+n)^3=A+B(n)+C(n)(n-1)+D(n)(n-1)(n-2)$$ And solve to find A,B,C,D.I know how to solve it. But I won't understand what it really mean and why he says ...
1
vote
1answer
54 views

Equation and roots finding without multiplying parenthesis

Today I'm studying some functions and I have found that equation: $f(x) = (x+1)(x+2)(x-3)$ I solve it by multiplying each parenthesis in order to have, after some addition, an equation like that: $...
1
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0answers
88 views

Exact Probability of reducibility of Bivariate Polynomials

I am considering polynomials of the form $$P(x,y)= \sum_{k=0}^n\sum_{l=0}^n a_{k,l}x^{k}y^{l}$$ where $n \in \mathbb{N}$. The coefficients $a_{k,l}$ are considered to be randomly generated from the ...
3
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5answers
88 views

Recommended books that discuss the Fundamental Theorem of Algebra?

I've been assigned to do a project on the Fundamental Theorem of Algebra and in particular discuss it's proofs and applications. I was wondering if anyone could recommend books that would aid me in my ...
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3answers
47 views

Series question involving a cubic polynomial

The question asks: Consider the polynomial $\displaystyle{\,\mathrm{f}\left(X\right) = X^{3} -6X^{2} + mX - 6}$, where $m$ is a real parameter. a. Show that: $\displaystyle{{1 \over x_{1}x_{2}} ...
10
votes
1answer
178 views

Bounding the “complexity” of irreducible factors of an integer polynomial

Given an integer polynomial $P(x) = a_0 + a_1 x + \cdots + a_n x^n$, there ought to be a reasonable bound on the "complexity" of its possible irreducible integer polynomial factors that allows us to ...
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votes
0answers
20 views

Zeros of a polynomial. [on hold]

If $F(i, x) $ is a polynomial where $i$ is a parameter and $\rho$ is the largest root of $F(0,x)$ and $F(i+1,x)\ge F(i, x)$, Prove that as $i$ increases $\rho$ will increase. I don't understand ...
1
vote
1answer
31 views

Finding the $\gcd$ of polynomials in $\Bbb R[x]$

Let $f(x)=6x^3-10x^2-6x+10$ and $g(x)=3x^2-14x+15$ in $\Bbb R[x]$. I want to find the $\gcd$ of these two polynomials. I am not really sure how to do this in general, but my approach was as follows: ...
1
vote
3answers
66 views

How to factorise $(x-1)^2 - (x-5)^2$

My attempt: $a = (x-1)$ $c = (x-5)$ $a^2 - c^2$ which is equal to: $$((x-1) - (x-5))((x-1)+(x-5))$$ But the correct answer is : $8(x-3)$ Can you explain, please?
1
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2answers
39 views

Getting characteristic polynomial from a small matrix

Sorry I don't know how to format matrices, but if I have this matrix $\pmatrix{1& 1& 0\\ 0& 0& 1\\ 1 &0& 1\\}$ How is the characteristic polynomial $λ^3 − 2λ^2 + λ − ...