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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Expression of coefficients of a product of Dirichlet polynomials

Suppose we have two Dirichlet polynomials: $$ f_1(s) = \sum_{n=1}^{m} \frac{a_n}{n^s} \\ f_2(s) = \sum_{n=1}^{m} \frac{b_n}{n^s} $$ Their product will also be a Dirichlet polynomial: $$ ...
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How to prove that $\,\,f\equiv 0,$ without using Weierstrass theorem?

Let $\,f:[0,1] \to \mathbb{R}$ continuous. Show that: If $$\int_0 ^1 x^k f(x)\, dx=0,$$ for all $k\in\mathbb N$, then $f\equiv 0$. I know that it can be proved using Weierstrass Theorem, ...
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If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?

I wanted to ask a separate question to focus on an elementary issue from my question Does the inverse of a polynomial matrix have polynomial growth?. Let $p : \mathbb{R}^n \to \mathbb{R}$ be a ...
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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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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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Chebyshev polynomials minimize the infinity-norm among all monic polynomials

Consider the monic Chebyshev polynomial $$\hat{C}_n(x) = 2^{1-n}\cos{(n\cos^{-1}{x})}.$$ Show, if $Q_n(x)$ is any other monic polynomial of degree $n$, that $$\left\|Q_n\right\|_\infty \ge ...
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When can variables simply be variables?

This may seem a somewhat strange question, but I've been tying myself in knots about it recently. When constructing a polynomial ring, you must formally define a polynomial as an ordered ω-tuple, ...
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Working with casus irreducibilis

I read about casus irreducibilis here. As an example of casus irreducibilis, it says we can factor $x^3 - 15x - 4$ to find $4$ as a root and it also has two other real roots. Using Cardano's method we ...
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Please Solve the following equation [closed]

Could you please help me to find the unkown power for the following equation : \begin{equation*} 3^x + 7^y + 5^z = 40,\\ 8^x + 6^y + 4^z = 30,\\ 5^x + 3^y + 9^z = 20. \end{equation*} Thanks. ...
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1answer
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Linearly Independent Linear Transformations

I am currently studying some theories of single linear transformations. I feels like I understant 99% of it, but there is still one thing that I have not been able to resolve. My book explains it by ...
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Derivatives and integrals of polynomials of two variables

Suppose I have a real-valued two dimensional polynomial $p(x,y)$ of order d. The partial derivatives inherit a nice structure, in particular knowing $\partial p/\partial x$ tells you $p$ up to the ...
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Are there no polynomials in $\mathbb{C[x]}:f^2 − Xf = −X^2 + 1$?

Are there no polynomials in $\mathbb{C[x]}:f^2 − Xf = −X^2 + 1$? What I did: $$ f^2 − Xf = −X^2 + 1 \iff f^2=Xf-X^2+1 $$ $\deg(f)=n \rightarrow \deg(f^2)=2n$, $\deg(Xf)=n+1$ and $\deg(-X^2+1)$=2 So ...
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Find the number of roots of a polynomial using Rouche's Theorem

Use Rouche's theorem to find the number of roots of the polynomial $z^5+3z^2+1$ in the anulus $1<|z|<2$. I am looking for a solution to this problem. My thoughts: This is a topic that ...
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4answers
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Finding horizontal tangents to a function.

Find the points at which the line tangent to the following function is horizontal $$q(x)=(x+3)^4(2x-1)^7$$ Every time I've gotten to the point of finding $x$ the numbers are all irrationally too ...
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Why is the zero polynomial not assigned a degree?

Yesterday, I read in my textbook, We assign degree to every polynomial and even a non-zero constant is assigned a degree $0$ but $0$ itself is not assigned a degree. Why is that? Why we don't ...
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Finding quadratic factors

Show that $(x-√3)$ and $(x+√3)$ are factors of $x^4+x^3-x^2-3x-6$. Hence write down one quadratic factor of $x^4+x^3-x^2-3x-6$, and find a second quadratic factor of this polynomial. My attempt: ...
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380 views

Irreducible factors of $X^p-1$ in $(\mathbb{Z}/q \mathbb{Z})[X]$

Is it possible to determine how many irreducible factors has $X^p-1$ in the polynomial ring $(\mathbb{Z}/q \mathbb{Z})[X]$ has and maybe even the degrees of the irreducible factors? (Here $p,q$ are ...
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1answer
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How to find all monomials $\left\{\left.x^n\in P_m\right|T(x^n)=0\right\}$ and which are in $\text{ker }T$?

Let $P_5$ be the set of one variable polynomials with real coefficients, whose degree are $\leq5$. Let $T$ be a linear transformation ...
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Polynomial with a prime number as a root

Is it possible to prove that this equation is false: $$ \sum_{i=0}^n a_i p^i = 0 $$ with following conditions: $a_i \in [-1;1]$; [Might $a\in\{-1,1\}$ have been intended here?] $p$ is a prime ...
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Polynomial maps on indeterminate of vector space of polynomials

I am studying polynomial rings and would like to get an idea of what it takes to study the problems of transforming polynomial forms by performing polynomial map on indeterminate of the polynomials. ...
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Find the possible values of a in the cubic equation.

Given that $(x-a)$ is a factor of $x^3-ax^2+2x^2-5x-3$, find the possible values of the constant $a$. I believe you first have to find the $a$ in the cubic equation then the other $a$ in $(x-a)$, but ...
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Let $p$ be a prime number $\ge2$ and $u = \cos\left(\frac 2p\pi\right)+i\sin\left(\frac2p\pi\right) \in \mathbb C$. Prove that …

Let $p$ be a prime number $\ge2$ and $u = \cos\left(\frac 2p\pi\right)+i\sin\left(\frac2p\pi\right) \in \mathbb C$. Prove that $u$ is root of $f(x)=x^{p-1}+x^{p}+...+x+1$. I know that $f(x)$ is ...
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Showing that an equation has a root in an interval

Show that the equation $x^4 - 7x^3 + 1 = 0$ has a root in the interval $[0,1]$. How would I go about working this out in steps?
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3answers
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Find $a\in\mathbb{R}$ such that $x_{1,2,3}\in\mathbb{Z}$

Consider $a\in\mathbb{R}$ and $x^3-x+a=0$ with $x_{1,2,3}\in\mathbb{C}$. We need to find $a\in\mathbb{R}$ such that $x_{1,2,3}\in\mathbb{Z}$. It seems be equivalent with to find a such that ...
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finding roots when polynomial does not equal zero

I was trying to solve this polynomial $$x(3-x^2)=1$$ I worked for the term $(3-x^2)$, I thought that this term cannot be $0$, thus $$3-x^2 >0$$ $x< \sqrt{3}$, $x<-\sqrt{3}$ is rejected ...
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Optimization of evaluation of polynomials with rational coefficients using algebraic constants

Considering it free to recall constants and already computed values, is there a univariate polynomial with rational coefficients that is easier to evaluate using constants that include irrational ...
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Writing $P_n=\sum_{\sigma \in \mathfrak{S}_n} X^{c_n(\sigma)}$ as irreducible factors in $\mathbb{Q}[X]$.

Let $\sigma \in \mathfrak{S}_n$, denote $\alpha_n(\sigma)$ the number of cycles in the decomposition of product of disjoint cycles. Let $$P_n=\sum_{\sigma \in \mathfrak{S}_n} ...
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Show that some of the root of the polynomial is not real.

\begin{equation*} p(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_3x^3+x^2+x+1. \end{equation*} All the coefficients are real. Show that some of the roots are not real. I don't have any idea how to do this, I ...
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root interlacing condition?

Consider the two functions $P(x)=e^x$ and $Q(x)=e^{-x}$, for $x\in\mathbb{C}$. Clearly, $P(x)$ and $Q(x)$ have no roots at all. Furthermore, it follows that $P(x)Q'(x)-P'(x)Q(x)=-2<0$, for any ...
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If two polynomials both of n degree have n identical real roots, are they equal? Proof?

CORRECTION: The polynomials don't have to be equal, but one has to be a constant multiple of the other. I ask the question because I saw this fact used in this solution to a problem: Problem: Given ...
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Find the values of the polynomial equation.

Find the values of A, B and C such that $A(x^2+4)+(x-2)(Bx+C)=7x^2-x+14$ My attempt to solve the question: $Ax^2+4A+Bx^2+Cx-2Bx-2C=7x^2-x+14$ $Ax^2+Bx^2+Cx-2Bx+4A-2C=7x^2-x+14$ ...
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Polynomial LongDivision

What would be the result of $x^3-4x^2-5$ divided by $x-3$ ? I am getting $4$ as my solution can someone prove me wrong, this is very confusing.
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Long division for multipolynomial expression, little o notation

I have this expression: $$\mathrm{Exp}=\frac{d^3(-12a^4)+d^2(4a^4-16a^3)+d(4a^3-6a^2-a)}{d^3(-12a^4+12a^3)+d^2(4a^4-20a^3+16a^2)+d(4a^3-11a+7a)+(1-2a+a^2)}$$ Is there any way I can take the second ...
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Product of linear factors

I have a polynomial $h=T^5+6T^4+6T^3+T+2$ in ring $\mathbb{F_7}[T]$. I should write it as a product of linear factors. So $h=(T+1)^2 (T^3+4T^2-3T+2)$. But $-3$ in $\mathbb{F_7}$ is $4$, so that ...
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Question about the K-Module $\cdot _\varphi : K[X]\times V \to V$

I managed to show several properties about the following mapping Let $K$ be a field, $V$ a finite dimensional $K$-Vectorspace and $\varphi \in \text{End}_K(V)$. $$ \cdot_\varphi : ...
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Number of Real roots of cubic

I just have a quick question about a polynomial and its roots, For example, I was solving the differential equation $$\frac{dy}{dx}=\frac{3x^2+4x+2}{2y-2}$$ I solved it using the basic methods of ...
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Show that the polynomial $x^8 -x^7 +x^2 -x +15$ has no real root. [duplicate]

Please check if my method is correct. Solution : Let $$f(x) = x^8 - x^7 + x^2 -x +15 $$ Now, let $g(x)= x^8 -x^7=x^7(x - 1)$ and $h(x)= x^2 -x=x(x-1)$. Thus, $$f(x)=g(x) + h(x) + 15$$ On analyzing ...
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interpolating and difference table, an old mid exam?!

For calculating divided (fraction) difference table for interpolating the points $(x_i, f_i)$, $i=1,2,...,n$; by using a polynomial with degree lower or equal to $n$, how many fraction was used? ...
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Prove or disprove: $p(x)$ diverges to infinity for $a_{n}>0$ [closed]

Prove or disprove that for any $n$ degree polynomial, $p(x)=a_{n}x^n+a_{n-1}x^{n-1}+a_{1}x+a_{0}$, if $a_{n}>0$, then $p(x)$ diverges to infinity as x tends to infinity. This is not homework.
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How to obtain a closed form for summation over polynomial ($\sum_{x=1}^n x^m$)? [duplicate]

What is the method for obtaining the polynomial equal to \begin{equation*} \sum^{n}_{x=1}x^m \end{equation*} for unknown $n$, and systematically for various values of $m$? I know it should be a ...
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4answers
530 views

Why do Z/7 have no cubic root of 2?

I was reading a textbook and came across the following line: Now we prove there is no cube root of 2 in $Z/7$. By noting that $(Z/7)^\times$ is cyclic of order 6, it will have only two third ...
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Ways to reduce polynomial modulus irreducible polynomial in F_2

Thank you all in advance for the help. I really appreciate your time in answering this question. I am trying to find a good summary of ways to reduce polynomial modules irreducible polynomial in ...
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Examples of rings whose polynomial rings have large dimension

If $A$ is a commutative ring with unity, then a fact proved in most commutative algebra textbooks is: $$\dim A + 1\leq\dim A[X] \leq 2\dim A + 1$$ Idea of proof: each prime of $A$ in a chain can ...
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Corollary of Gauss's Lemma (polynomials)

I am trying to prove the following result. I have outlined my attempt at a proof but I get stuck. Any help would be welcome! Theorem: Let $R$ be a UFD and let $K$ be its field of fractions. ...
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Does $o(|x-a|^n)$ approximation by a polynomial imply existence of derivatives?

While reviewing the topic of Taylor expansion, I've noticed that while in all statements about the $n$th order Taylor polynomial of $f:\mathbb R \to \mathbb R $, it's always assumed that $f\in C^n$, ...
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1answer
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How many monic primitive quadratic polynomials are there in $Z_{7}[x]$?

A theorem states that "for each prime p and for each integer $n \ge 1$, there exists a monic irreducible polynomial of degree n in $Z_{p}[x]$". I am not sure if this theorem will help answer my ...
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Proving that the ideal of polynomials in $F[x,y]$ with $0$ constant coefficient is not principal

Prompt: Let $F[x,y]$ denote the domain of all the polynomials $\sum a_{ij}x^{i}y^{j}$ in two letters $x$ and $y$, with coefficients in $F$, where $F$ is a field. Let $J$ be the ideal of $F[x,y]$ which ...
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I need help with polynomial long division

When proving $2^n - 1$ is composite if $n$ is composite this product $(x^a-1)(x^{(a-1)b} + x^{(a-2)b} + x^{(a-3)b} + ... + x^a + 1) = x^{ab} - 1$ comes up. I am not sure how to verify this by long ...
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4answers
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Find the roots of the polynomial? (Cardano's Method)

$y^3-\frac7{12}y-\frac7{216}$ This is part of Cardano's method, so I've gotten my first root to be: $y_1=\sqrt[3]{\frac7{432}+i\sqrt{\frac{49}{6912}}}+\sqrt[3]{\frac7{432}-i\sqrt{\frac{49}{6912}}}$ ...
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1answer
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Explicit formula for interpolating polynomial

$a\in(0,1)$ is fixed. $M\in\Bbb Z_{>1}$ is fixed. What is $f(x)$ given that $$f(0)=0\mbox{, }f(M)=1+a\mbox{, }f(1)=1-a$$$$\mbox{ }f(x)\in(1-a,1+a)\mbox{, }\forall x\in(1,M)?$$ What is $g(x)$ ...