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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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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3answers
22 views

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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0answers
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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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2answers
22 views

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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2answers
19 views

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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1answer
18 views

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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60 views

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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5answers
369 views

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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1answer
15 views

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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2answers
36 views

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 non-rational ...
2
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1answer
23 views

Prove or disprove: $p(x)$ diverges to infinity for $a_{n}>0$ [on hold]

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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2answers
16 views

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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2answers
44 views

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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1answer
20 views

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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0answers
17 views

Solving Quartic equation by Descartes method [on hold]

Please give me Quartic equation which can be solved by Descartes method and please solve it by Descartes method Thanks
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1answer
23 views

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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4answers
521 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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1answer
20 views

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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1answer
47 views

Degree of a polynomial? [on hold]

Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...
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1answer
17 views

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 $g(x)$ ...
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4answers
43 views

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
31 views

How to approximate Heaviside function by polynomial

I have a Heaviside smooth function that defined as $$H_{\epsilon}=\frac {1}{2} [1+\frac {2}{\pi} \arctan(\frac {x}{\epsilon})]$$ I want to use polynominal to approximate the Heaviside function. ...
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2answers
40 views

Cyclic Equation. Prove that: $\small\frac { a^2(b-c)^3 + b^2(c-a)^3 + c^2(a-b)^3 }{ (a-b)(b-c)(c-a) } = ab + bc + ca$?

This is how far I got without using polynomial division: \begin{align} \tiny \frac { a^{ 2 }(b-c)^{ 3 }+b^{ 2 }(c-a)^{ 3 }+c^{ 2 }(a-b)^{ 3 } }{ (a-b)(b-c)(c-a) } &\tiny=\frac { { a }^{ 2 }\{ { b ...
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2answers
47 views

a question about abstract algebra,the order of $\Bbb Z_{5}[x]/ (x^3+x+1)$

Firstly, I have proven that $x^3+x+1$ is irreducible in $\Bbb Z_{5}[x]$,then how can I know the order of $$\Bbb Z_{5}[x]/ (x^3+x+1),$$ where $(x^3+x+1)$ means the ideal generated by $x^3+x+1$. Can ...
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1answer
60 views

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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1answer
22 views

Order approximation for rational polynomial

I have this fraction: $\frac{(-12a^3)d^3 + (4wa^3 - 16a^2)d^2 + (5wa^2 - 8a)d - a^2w^2 + 2aw - 1}{(- 12wa^4 + 12a^3)d^3 + (4a^4w^2 - 20a^3w + 16a^2)d^2 + (4a^3w^2 - 11a^2w + 7a)d + a^2w^2 - 2aw + 1}$ ...
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0answers
9 views

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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2answers
91 views

What wolfram does to factor $x^6+x^2+2$?

I am learning polynomials and I am trying to understand what wolfram did to obtain $$(x^2+1)(x^4-x^2+2)$$ from $$x^6+x^2+2$$ It does not show me the step-by-step option in this case and I got ...
4
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1answer
56 views

For a polynomial $f\in K[x]$, when is there a constant $c\in K$ such that $f+c$ is irreducible?

I was working on a different problem when the following question occurred to me: For a polynomial $f\in K[x]$, is there always a constant $c\in K$ such that $f+c$ is irreducible? Obviously this is ...
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2answers
56 views

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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0answers
17 views

How many kind of basis function to approximate an arbitrary function

I am finding a list algorithm to approximate an arbitrary function. Such as Bernstein, he said that a linear combination of Bernstein basis polynomials $$B_n(x) = \sum_{\nu=0}^{n} \beta_{\nu} ...
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0answers
78 views

How to make addition of two polynomials have no integer root.

Consider I have two degree $d$ polynomials $f_1$ and $f_2$, and they do not have any root in common. I need to compute $f_3=f_1+f_2$, but $f_3$ may have some roots in $R$. So I pick two random ...
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0answers
22 views

How to approximate a trigonometric to make less computation complexity

I having a trigonometric function such as $$ p_2(s) = \begin{cases} \frac {1}{(2 \pi)^2}(1-\cos (2 \pi s)), & \text{if $s \le1$ } \\ \frac {1}{2 }(s-1)^2, & \text{if $s >1$ } ...
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0answers
19 views

Are the conditions ${c=ae+(b-e), d=e(b-e)}$ and $ad=(b-e)[c-(b-e)]$ equivalent?

Respected All. Please help me on the following. Suppose that $x^5+ax^4+bx^3+cx^2+dx+e\in \mathbb R[x]$ can be written as $$(x^3+a_1x^2+b_1x+1)(x^2+b_2)$$ then we must have \begin{align*} ...
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1answer
27 views

Finding the matrix representation of a linear transformation $ T: P_{3} \to \text{M}_{2 \times 2} $.

Define a function $ T: P_{3} \to \text{M}_{2 \times 2} $ by $$ T \! \left( a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} \right) = \begin{pmatrix} a_{3} & a_{0} \\ a_{2} & a_{1} \end{pmatrix}. ...
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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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2answers
20 views

Polynomials of degree less than $n$ that agree at $n$ values

Suppose $\deg(a(x))$ and $\deg(b(x))$ are both less than $n$. If $a(c) = b(c)$ for $n$ values of $c$, prove that $a(x) = b(x)$. This seems simple, since if the $a(c) = b(c)$ for $n$ values of ...
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2answers
30 views

Lowest degree polynomial satisfying $\ P(n)=1/n$ for first n natural numbers

So say I wanted to find the lowest degree polynomial satisfying$\ P(1)=1, P(2)=1/2, P(3)=1/3$. Is there some sort of formula where I can put in$\ n$ and there will be a polynomial with coefficients in ...
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1answer
38 views

Find the roots of the polynomial in $Z_5$

The polynomial is: $2x^{219} + 3x^{74} + 2x^{57} + 3x^{44}$. Find the zeros. Now my first step, which I believe shall be correct is to reduce the exponents of the polynomial in mod 5. Thus: 219 ...
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1answer
23 views

Vanishing set of irreducible polynomials

Question: Find irreducible $f,g \in \mathbb{R}[x,y]$ such that $V(f) = V(g) \neq 0$ with the added requirement $f \neq \lambda g$ for $\lambda \in \mathbb{R} - \{0\}$. Attempt: I think $f(x,y) = x^2 ...
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1answer
21 views

Solve polynomial equation in $\mathbb{C}[x]$

Find the polynomials $f,g \in \mathbb{C}[x]$ with complex coefficients such that: $$f(f(x))-g(g(x))=1+i,\\f(g(x))-g(f(x))=1-i$$ for all $x\in\mathbb{C}$. I think I have this problem almost ...
2
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0answers
35 views

Assume that d,u,q are all positive integers. Also, d-u is divisible by q, u-q is divisible by d and q-d is divisible by u.

Assume that d,u,q are all positive integers. Also, d-u is divisible by q, u-q is divisible by d and q-d is divisible by u. What is the solution for $d,u,q$? Am I right to assume that the solution is ...
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0answers
38 views

Observation about Polynomials Addition

For two polynomials $f_1=(x-a)(x-b), f_2=(x-a)(x-e)$, if we add them together: $f_3=f_1+f_2$, $f_3$ only has an integer root that is $a$. I observed that it'd possible to make $f_3$ have more than one ...
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1answer
41 views

Sums of two irreducible polynomials over $\mathbb{Z}$

Please help me to prove that any polynomial with integer coefficients can be represented as a sum of two irreducible polynomials over the ring $\mathbb{Z}$.
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0answers
17 views

Bound on roots of a monic polynomial

Let $p(x)$ be a real monic polynomial of degree $n$. The following information is given about $p(x)=0$: all roots are real there is no zero root exactly $k$ roots are positive and $n-k$ roots are ...
2
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1answer
69 views

Prove $p(x)>0$ for $x>b$

This is a question from a past paper which I have no solution to. Let $p(x)=x^n + a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}, n\geq 1$ be a polynomial of dgree n and let $b=1+|a_{1}|+\cdots ...
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1answer
9 views

quadratic form polynomial divisibility vs. matrix pointwise multiplication. [on hold]

Given matrix $V',W',Y'$ is of $d\times m (d\le m)$ ; column vector $c$ is of size $m$; $r_i, i=1,...,d$ are distinct; and each row of the matrix A is $A_i=(r_i^0 ... r_i^{d-1})$. So, A is of $d\times ...
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2answers
38 views

Checking irreducibility of a polynomial over a finite field

A part of a coding theory course I am doing includes some questions on irreducible polynomials. I have a question with solution but am worried I have interpreted it incorrectly. So for $\mathbb F_5$ ...
4
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0answers
55 views

Find all positive values for j,k,l such that j, k, l are positive integers and (j-k)|l, (k-l)|j, (l-j)|k.

Find all possible values of $j,k,l$ such that $j, k, l$ are positive integers and $(j-k)|l, (k-l)|j, (l-j)|k$. As I understand that using divisibility properties, it is possible to come to some ...
3
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2answers
52 views

Find a second root of $x^3+px+q$ given the first root

This is a problem from Artin where given one root $a$, you have to find an equation for a second root in terms of $a$, $p$, $q$, and the square root of the discriminant $\delta$. Here's what I have ...