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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131 views

What is the relationship between saying “a Taylor series converges for all $x$” and “a Taylor series converges to a function, f(x)”

Given the following Taylor series: $1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}- \dots$ We know that: It converges for all of $x$ It converges to the function $\cos x$ The ...
2
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0answers
74 views

About the condition such that $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ is the $n$-th power of an integer for every integer $x$

Question : Is the following true for any $n\ge 2\in\mathbb N$? Letting $a_n, a_{n-1}, \cdots, a_1,a_0\in\mathbb R$ be constants, the necessary and sufficient condition for $a_n, a_{n-1}, ...
1
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1answer
91 views

Real Polynomials in inner product space projections?

Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Define $$ \langle p,q\rangle = p(1)q(1)+p'(1)q'(1)+p''(1)q''(1)+p'''(1)q'''(1). $$ (i) Prove ...
1
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1answer
60 views

Find a basis for $p \in P_2(\mathbb R) $ with $p(7) = 0$?

Find a basis for the subset: $$ S = \{\;p \in P_2(\mathbb R)\;\; |\;\; p(7) = 0\; \} $$ I'm not sure how to approach this question. $$ p(7) = a_0 + 7a_1 + 49a_2 = 0 $$ $$ a_0 = -7a_1 - 49a_2 $$ $$ ...
7
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1answer
451 views

Find $f(5)$ of a non-constant polynomial function $f(x)$

Suppose $f(x)$ is a non-constant polynomial such that $f(x^ 3) − f(x ^ 3 − 2) = f( x )\cdot f(x) + 12$ for all $x$. Find $f(5)$? I find this problem on Quora just now, and I try to solve it but do ...
8
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2answers
640 views

closed-form expression for roots of a polynomial

It is often said colloquially that the roots of a general polynomial of degree $5$ or higher have "no closed-form formula," but the Abel-Ruffini theorem only proves nonexistence of algebraic ...
2
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1answer
47 views

Question about a polynomial's degree

How can we show that if $p(x)$ is a polynomial of degree $d-1$, then $$\sum_{k=n_0}^n p(k)$$ is a polynomial in $n$ of degree $d$?
2
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1answer
81 views

Gröbner bases: Polynomial equations. Solution $x$ to $G \cap k[x_1, .., x_i]$ imply solution to $G \cap k[x_1, .., x_i, x_{i+1}]$, $x$ plugged in.

I'm have been studying Gröbner bases for a while now and seen a few examples in my textbook / exercises. Let $\mathcal k$ be a field and $\mathcal k[x_1,..,x_n]$ a polynomial ring. I wish to solve a ...
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2answers
59 views

Are these ideals the same?

I have already proved that $(X^3-Y^3,X^2Y-X)\subseteq(X^2-Y,X-Y^2)$ since the elements $X^3-Y^3$ and $X^2Y-X $ can be written as a linear combination of $(X^2-Y,X-Y^2)$. However, I can't write ...
0
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1answer
28 views

Polynomial as a sum of an exponent, -1 and another polynomial

This is from the IMOmath website: Denote $P(x)=(1+x)(2+x)…(p−1+x)$. We know that $P(x)=x^{p-1}−1+pQ(x)$ fo-r some polynomial Q(x) with integer coefficients. (p is prime) How do we 'know ...
2
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0answers
73 views

For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$?

For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$? In this paper on quantum computing (last paragraph of page 25), Dorit ...
0
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1answer
96 views

Inconsistent system of simultaneous equations

Let $F$ be an algebraically closed field, and $f_1,\ldots,f_n$ polynomials in $k$ variables over $F$. The system of simultaneous equations $$\mathcal{F}: ...
2
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1answer
116 views

How to verbalize $R[x]$?

Let $R$ be a ring, and let $x$ be an indeterminate. Let $R[x]$ denote the ring of polynomials in $x$ with co-efficients in $R$. How to most efficiently read (i.e. pronounce) the symbol $R[x]$ while ...
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2answers
929 views

Factoring multivariate polynomial

I'm trying to factor $$x^3+x^2y-x^2+2xy+y^2-2x-2y \in \mathbb{Q}[x,y].$$ The hint for the exercise is to use the recursive multivariate polynomial form. So I'm using $\mathbb{Q}[x][y]$: $$ x^3 + ...
2
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0answers
61 views

Approximating polynomial of higher degree

Suppose that we approximate a function $f(x)$ for $x$ near $0$ by a polynomial of degree $n$: $$f(x)\approx P_n(x)=C_0+C_1x+C_xx^2 + \dots + C_{n-1}x^{n-1} +C_nx^n$$ We need to find the values ...
0
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1answer
80 views

Finding a generating function for a pattern

I was working on this projecteuler.com problem, and I was very interested by the premise. Essentially, given n terms, find an ...
2
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1answer
154 views

Trouble solving polynomial equation with exponent

I'm having trouble solving this equation.It looks simple, but I just can't find the answer.Can someone help me? $$9x^4-13x^2+4 = 0$$
2
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1answer
59 views

Zeros of polynomials are continuous

For two sets $A,B$, let $d(A,B)=\sup_{x\in A}\inf_{y\in B}|x-y|+\sup_{y\in B}\inf_{x\in A}|x-y|$. Let $p(z)=a_nz^n+\ldots+a_0$, and let $\epsilon>0$. Show that there exists $\delta>0$ such ...
5
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2answers
465 views

Is there something like Cardano's method for a SOLVABLE quintic.

So there is no quadratic formula equivalent for a GENERAL fifth degree equation, but is there an equivalent formula for a SOLVABLE fifth degree equation.
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2answers
68 views

Finding the roots of 4096x^3-10496x^2+152576x - 961=0 (1 root and 2 complex)?

I don't know how to find the roots of 4096x^3-10496x^2+152576x - 961=0 I try using wolfram and http://en.wikipedia.org/wiki/Cubic_function. I don't really understand it can someone please explain how ...
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2answers
104 views

How to solve a 4th degree polynomial?

I am feeling difficulty to find the roots of this 4th degree polynomial: $3x^4+26x^3+77x^2+84x+24=0$ Factorization methods have been tried.
4
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1answer
93 views

Approximating $e^{inx}$ by polynomials

Show that every function $e^{inx}$ can be uniformly approximated on $[-\pi,\pi]$ by polynomials in $x$. Using the power series expansion, ...
0
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0answers
132 views

express as a product of three factors $ x^4 + 3x^3 +4x^2 -6x -12 $

express $ x^4 + 3x^3 +4x^2 -6x -12 $ as a product of three factors i can't do it by means of synthetic division the factors are probably not integers but how else am i supposed to simplify it? i ...
0
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2answers
75 views

Solving 4th degree polynomials

I am feeling difficulty to find the roots of this 4th degree polynomial: $$2x^4+18x^3+58x^2+72x+24=0$$ Factorization methods have been tried.
2
votes
3answers
301 views

$x^6+x^3+1$ is irreducible over $\mathbb{Q}$

I have been trying to prove that $x^6+x^3+1$ is irreducible over $\mathbb{Q}$ (or $\mathbb{Z}$ since by Gauss' Lemma is the same), but I can't. Any idea of how to do so?
0
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2answers
64 views

roots of polynomial equation

How to find the roots of $x^5-2^5$ by hand. I see that we get a root of $x=2$ and 4 complex roots (should come in pairs). Not sure how to work out the complex roots. Do we need to convert to polar? ...
0
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1answer
74 views

Prove that the elements $2x$ and $x^2$ have no LCM in the ring of integral polynomials with even coefficient of $x$

Let $A$ be the subring of $\Bbb Z[x]$ consisting of all polynomials with even coefficient of $x$. Prove that the elements $2x$ and $x^2$ have no lowest common multiple. Hints please!
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1answer
388 views

How solve this equation

Find the equation $$x^5+10x^3+20x-4=0$$ My try:I think this equation maybe take Trigonometric functions Now I have solution:let $x=t-\dfrac{2}{t}$,then ...
0
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0answers
132 views

When two polynomials of several variables are equal

Let $P(X_1,\ldots,X_n)=Q(X_1,\ldots,X_n)$ be two polynomials with rational coefficients such that total degree of P and Q are same and $P(m_1,\ldots,m_n)=Q(m_1,\ldots,m_n)$ for all large ...
0
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1answer
408 views

Find a basis for ker(T) and range(T) for the given transformation and compute T(5x-4)

I am not really having trouble with $a)$, $c)$, $d)$ or $e)$. For $a)$ I put that $B$ is a basis for $\mathbb{P}^1$ because it has ${\rm dim} = 2$ and the highest degree is 1, and for $B'$ it has ...
0
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2answers
442 views

Continuous real valued functions and inner product space?

Let $V$ be the space of all continuous real valued functions on the interval $[1,4]$ with the inner product defined by: $$\langle f,g\rangle = \int_1^{4} f(t)g(t)\,dt.$$ (i) Find an ...
0
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1answer
48 views

Sketching fuctions - standard meaning of terms?

When sketching the graphs of polynomials, do certain terms always have the same meaning? For example, in $f(x) = x^n + c$, the intersection with the y-axis is always $c$, no matter how high the ...
2
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1answer
336 views

Sum of multinomial coefficients with constraints

The title doesn't reflect the question properly, since I don't know enough about combinatorics to get it right, here. Feel free to change the title. From the multinomial theorem, we can deduce, that ...
3
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3answers
11k views

Basis of the polynomial vector space

I doesn't understand how to find a basis for a polynomial vector space. Can someone help me with an example?
3
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2answers
98 views

Solve diophantine equation $x^2 - 2y^2 = x - 2y$

Thanks to internet, I found and understand how to solve diophantine $x^2 - Dy^2 = 1$. Now I would like to solve the following diophantine equation : $$x^2 - 2y^2 = x - 2y$$ but I don't know how to do ...
2
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1answer
53 views

Let $F$ be a field and $f \in F[x]$. Prove that $\left\{ g(f(x)):g\in F[x]\right\}$ is equal to $F[x]$ if and only if $deg(f)=1$.

I am trying to solve the following exercise Let $F$ be a field and $f \in F[x]$. Prove that $\left\{ g(f(x)):g\in F[x]\right\}$ is equal to $F[x]$ if and only if $deg(f)=1$. I have only managed ...
3
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1answer
49 views

Different real roots polynomial, roots of $P'+aP$

Let $P$ be a polynomial of degree $n$ with real roots $t_1<t_2\ldots<t_n$. Show that $P' + aP$, with $a\in\mathbb{R}$, has only real roots. Is easy to conclude if $a=0$, by Rolle's theorem. ...
2
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2answers
45 views

Prove inequality for polynomials

Let $a_0,a_1,\dotsc,a_n \in \mathbb C$ and $p(z) = a_0+a_1z+\dotso+a_nz^n$. How can one show that $\lvert p(z)\rvert \ge \lvert a_n\rvert\lvert z\rvert^n-\big\lvert\sum_{j=0}^{n-1}{a_jz^j}\big\rvert$ ...
2
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1answer
33 views

Additivity of polynomial functions in characteristic $p$

Suppose $k$ is a field of characteristic $p$, and $f\in k[x]$ such that $f(a+b)=f(a)+f(b)$ for all $a,b\in k$. Does it follow that $f(x)=\sum c_ix^{p^i}$ for some $c_i\in k$? I'm fairly certain this ...
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1answer
91 views

Matrix polynomial factorization

This is about exercise 1207 from the book "Problems and Solutions in Mathematics", 2nd edition, by Ta-Tsien. Let $p$ be a prime and let $V$ be an $n$-dimensional vector space over the finite field ...
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0answers
120 views

How to factorize an equation?

How can I quickly factorize an equation without using Ruffini's rule or polynomial division? Take this equation as an example: x^4-9x^2+20x=0
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1answer
59 views

How find this value of $\prod_{1\le i<j\le n}(w^i-w^j)^2$

give the positive integer number $n$, and $w=\cos{\dfrac{2\pi}{n}}+i\sin{\dfrac{2\pi}{n}}$ where $i^2=-1$ find the vaule $$\prod_{1\le i<j\le n}(w^i-w^j)^2$$ My try:note $$w^n=1$$ ...
2
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3answers
101 views

Finding element in binary representation of $GF(2^6)$

I got the following task: Let $F = GF(2^6 )$ be K[x] modulo the primitive polynomial $h(x) = 1 +x ^2 +x ^3 +x ^5 +x ^6$ , and let $\alpha$ be the class of x. I have a table with the binary ...
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0answers
34 views

Cyclic Code Galois

I have a generator polynomial over GF(3) for a cyclic code. What is the fastest way to find the minimum distance of this code?
3
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1answer
73 views

Mutual dependency of polynomial expressions

Suppose you are given the values of $m$ polynomial expressions in $n$ variables. That is we know that $P_1(x_1,x_2,...,x_n)=a_1,P_2(x_1,x_2,...,x_n)=a_2,...,P_m(x_1,x_2,...,x_n)=a_m$ for some ...
3
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1answer
101 views

Barbeau's Polynomials, Exploration 2

I'm reading Barbeau's Polynomials. Let $m$ be a positive integer. It is a remarkable fact that the numbers from $1$ to $2^{m+1}$ inclusive can be subdivided into two subsets $A$ and $B$ such that, ...
26
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3answers
444 views

Patterns of the zeros of the Faulhaber polynomials (modified)

Faulhaber polynomial of order $p \in \Bbb{N}$ is defined as the unique polynomial of degree $p+1$ satisfying $$ S_{p}(n) = \sum_{k=1}^{n} k^p $$ for $n = 1, 2, 3, \cdots$. For example, ...
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1answer
99 views

Problem involving trigonometry and cubics

One of my teachers proposed me the following problem: $$\text{If } (3\sec x+\csc x)\sin x=5\cos^2 x\text{, calculate } z=\tan x+\sec x$$ I started by manipulating $$3\tan x +1=5\cos^2 x$$ $$\sec^2 ...
1
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1answer
127 views

Primitive polynomial

Prove that $x^5 + x^2 + 1$ is a primitive polynomial over ${\mathbb F}_2$. I have already proved that the above polynomial is irreducible. Do I have to exhaustively prove that the above polynomial ...
0
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1answer
68 views

Let $f(x) \in F[x]$ and assume that $f(x)|g(x)$ for every nonconstant $g(x) \in F[x]$. Show that $f(x)$ is constant

because $f(x)|g(x)$ $f(x)$ must share at least one root with $g(x)$ and because $g(x)$ could be any degree polynomial that don't share the same roots (ex: $x+1$, $x-2$) $f(x)$ must be a constant ...