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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2answers
18 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
37 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
37 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
0answers
14 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 ...
1
vote
1answer
18 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. *...
-4
votes
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, ...
3
votes
1answer
80 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'...
0
votes
5answers
174 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 ...
1
vote
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 ...
1
vote
2answers
57 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 ...
1
vote
0answers
34 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 ...
4
votes
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 $\mathbb{Q}(i)$ ...
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
0answers
10 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 ...
8
votes
5answers
125 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.
0
votes
1answer
49 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
votes
1answer
34 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 ...
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 ...
1
vote
3answers
48 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 ...
2
votes
3answers
58 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(...
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^...
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 ...
1
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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 ...
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 ...
5
votes
1answer
44 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 $\mathbb{Z}/p\mathbb{Z}$?
1
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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
votes
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: $...
0
votes
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}} ...
-4
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
30 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 − ...
0
votes
2answers
50 views

What is the coefficient of $x^{2m}$ in $(1 + 4 x - 2 x^2 + 4 x^3 + x^4)^m$?

For each positive integer $m$, write $(1 + 4 x - 2 x^2 + 4 x^3 + x^4)^m = \sum_{j = 0}^{4m} b_j^{(m)} x^j$. What is $b_{2m}^{(m)}$ in terms of $m$?
0
votes
0answers
24 views

When is $\{ X^{mk} \ : \ 0 \leq k \leq n-1\} $ a basis for $R[X]/(f)$?

Let $R$ be a commutative ring and $f \in R[X]$ irreducible with degree $n$. Let $m$ be an integer such that $0 \leq m \leq n-1$. Can we say that $$ \mathcal{B}\ := \ \{ X^{mk} \ : \ 0 \leq k \leq n-1\...
0
votes
2answers
22 views

Need explanation on the working

I'm a bit confused on the working to this question. A friend of mine gave me a solution but I dont quite understand why is the working like that. Below is the question: 'When P(x) is divided by (x-1) ...
4
votes
2answers
49 views

If $ax^2 +bxy+cy^2+5x-2y+3$ divided by $x-y+1$ has remainder $0$, determine $a$, $b$, and $c$. [closed]

If $ax^2 +bxy+cy^2+5x-2y+3$ divided by $x-y+1$ has remainder $0$, determine $a$, $b$, and $c$. I do not know how to approach this problem and would appreciate advice how to proceed.
0
votes
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
vote
0answers
17 views

Required polynomial order for 2D least square function fit

I am working with a point cloud of approximately 500 points which has the form $p = f(x,y)$ and I need to find a function $\hat{f}$ that will correctly approximate $f$ on all of its domain. To do so, ...
0
votes
0answers
17 views

How to prove in $r_1p_1 +r_2p_2 =u\gcd(p_1,p_2)$, $u$ is a uniformly random polynomial.

Hypothesis: All polynomials are defined over a finite field $\mathbb{F}_p$, where $p$ is a large prime number (e.g. 128-bit prime number). Assume we have two fixed polynomials $p_1$ and $p_2$ of ...
0
votes
2answers
66 views

To show that $n!f(x) \in \mathbb{Z}[x]$

If $f(x)$ is a polynomial such that if $y \in\mathbb{Z}$, then $f(y)\in\mathbb{Z}$, show that there exist $n$ such that $n!f(x)\in\mathbb{Z}[x]$
1
vote
0answers
35 views

Probability question, can I reset the window or not

There is a wall street banker. The banker invests in a kind of share called as options. The main features of this share is as follows: You make a bet with a specified amount of information as to ...
0
votes
1answer
49 views
0
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3answers
67 views

Finding the root of an indefinite polynomial

$0 = (a-n) x^{n-1} + ax^{n-2} + ax^{n-3} + \cdots + ax + a$ What is $x$ in terms of $a$ and $n$? I don't even know what this form of polynomial is called.
6
votes
2answers
280 views

Ways to find irrational roots of an n degree polynomial

I am trying to write a program to find the roots a given polynomial of degree N, with the form $$ A_{0}X^{N}+A_{1}X^{N-1}+A_{2}X^{N-2}+A_{3}X^{N-3}+...+A_{N} $$ I know that if there are rational ...
1
vote
2answers
39 views

How to prove $\prod_{i=1}^{n}(x-4i+2)(x-4i+1)>\prod_{i=1}^{n}(x-4i+3)(x-4i)$ for all $x\in\mathbb{R}$?

I would like to prove that for $n\in\mathbb{N}$ we have $f_n(x):=\prod_{r=1}^{n}(x-4r+2)(x-4r+1)>\prod_{r=1}^{n}(x-4r+3)(x-4r)=:g_n(x)$ for all $x\in\mathbb{R}$ (actually it would suffice for $n$ ...
15
votes
2answers
465 views

Sum of roots rational but product irrational

Suppose that $x_1,x_2,x_3,x_4$ are the real roots of a polynomial with integer coefficients of degree $4$, and $x_1+x_2$ is rational while $x_1x_2$ is irrational. Is it necessary that $x_1+x_2=x_3+x_4$...
0
votes
2answers
42 views

Number of real roots of $\frac{a_1}{a_1-x}+\frac{a_2}{a_2-x}+…+\frac{a_n}{a_n-x}=2016$ for $0<a_1<…<a_n$?

Does it have exactly $n$ roots? Would replacing the R.H.S. of the equation with any other real number change the outcome? I can show that the equation has no complex roots. But how to find the number ...