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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Partial Fraction Decomposition of a Polynomial division

Question :Write $$\frac{x^5}{(x^2+1 )(x+1)^2}$$ as a sum of partial fraction What I've tried is to do polynomial long division twice to reduce the degree of numerator to be smaller than denominator ...
2
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1answer
18 views

Possible division by zero when repeatedly factoring and canceling (x-r) and evaluating the resulting polynomial at x=r…

The following proof is from 'Two Proofs of the Existence and Uniqueness of the Partial Fraction Decomposition', page 4. Lemma 2.1. Let $h(x), g(x) \in R[x]$ (real polynomials), $k$ a positive ...
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3answers
218 views

Factors in a cubic equation

I have no idea how to go about this. Any Hint? Suppose that $(x-3)$ is a factor of $$kx^3 - 6x^2 + 2kx - 12.$$ Solve for $k$.
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0answers
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Why is polynomial convolution equivalent to multiplication in F[x]/(xn−1)?

Why is polynomial convolution equivalent to multiplication in $F[x]/(x^n-1)$? From this, I still can not understand how to get this $$ \begin{align} &f*g ...
2
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2answers
80 views

Polynomial in several variables over $GF(2)$

Can anyone please explain how this Lemma has been proved? Lemma: Let $f$ be a nonzero polynomial in variables $x_1,\ldots,x_n$ over $GF(2)$, and let $d$ be the maximum degree of $f$ with respect to ...
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1answer
30 views

If $P(x)$ and $Q(x)$ are both factors of $H(x)$

$P(x)$ and $Q(x)$ are both quadratic polynomials and both are factors of a cubic polynomial $H(x)$ such that: $$H(x) = (x - a)P(x) \space \text{AND} \space H(x) = (x - b)Q(x)$$ For distinct $a,b$ ...
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0answers
15 views

Under what condition is substitution possible?

Let $R$ be a ring and $f_1,...,f_n$ be in the center of $R[X_1,...,X_n]$. Then, we can always have a ring endomorphism over $R$ which sends $X_i$ to $f_i$. Likewise, I'm curious under what consition ...
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1answer
24 views

What is a formal model for equations in non-commutative ring?

The standard formal modeling for polynomials is the polynomial ring $R[X_1,...,X_n]$ which is a monoid ring $R[\mathbb{N}^n]$ over an rng $R$. Under this construction, it is possible to commute ...
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2answers
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$5x\big(1+\frac{1}{x^2 +y^2}\big)=12$ ; $5y\big(1-\frac{1}{x^2 +y^2}\big)=4$ find $x$ and $y$

I already tried to solve using substitution and cross multiplication method . I got the first simplified (1)$$\frac{12}{5x}=1+ \frac{1}{x^2} +y^2$$ (2) $$\frac{4}{5y}=1- \frac{1}{x^2+y^2}$$ Adding ...
3
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1answer
65 views

Properties of distribution of zeros of polynomial

Polynomial $p_n(z) = (1 + \frac{z}{n})^n - 1$ has a property that all its zeros lie on the circle of radius $n$. It is easy to see because $$\frac{z}{n} = e^{\frac{i2\pi k}{n}} - 1$$ So we can "fit" ...
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1answer
17 views

Ideas and methods in deciding solvability of rational expression equals integer

I would like to know if there are results concerning the solvability - or even the solution - of equations of the form $$ R(t)=z, $$ where $t$ and $z$ are both unknown, $t\in \mathbb{Q}$, ...
2
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2answers
26 views

Can the concept of congruence be applied to the remainder of a polynomial division?

I know this is a very simple question, so please I apologize but I am not familiar with it: Can the concept of (modular arithmetic) congruence be applied to the remainder of a polynomial ...
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0answers
46 views

Expansion of $(6 + 3x + x^2)^n$

In the role playing game Exalted, there is a dice mechanic whereby you have a certain number of 10-sided dice in a dice pool and when you roll them, each die showing a 7, 8, or 9 count as one success ...
2
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1answer
13 views

Dimension of set of all homogeneous polynomial of degree $d$ in $n$-variables over a field $F$

Let, $V$ be a set of all homogeneous polynomial of degree $d$ in $n$-variables over a field $F$. Then dimension of $V_F$ is (A) $\left(\begin{matrix}n\\d\end{matrix}\right)$ (B) ...
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2answers
34 views

Finding value of a polynomial

If it is given that f(x) is a 5th degree polynomial such that $\ f(1)=1;f(2)=3;f(3)=5;f(4)=7;f(5)=9$ what is the value of f(6)? I usually solve these kind of problems by making equations but no luck ...
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1answer
70 views

Polynomial between $0$ and $1$ that produces largest integral

Question: Let $n\in \mathbb{N}$. Find the polynomial $p(x) = \sum_{i=1}^n a_ix^i$ that satisfies $p(1) = 1$ (and $p(0)=0$ since we already have $a_0=0$) $p(x) \in [0,1]$ for all $x\in ...
2
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2answers
48 views

Show that $f=x^3+7x+5$ has no roots in $\mathbb {Q}(\sqrt[4]{2})$

Show that $f=x^3+7x+5$ has no roots in $\mathbb {Q}(\sqrt[4]{2})$. I'm given a hint: suppose $\alpha$ is a root of $f=x^3+7x+5$ and $\alpha\in\mathbb{Q}(\sqrt[4]{2})$, compute ...
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5answers
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Remainder of $(1+x)^{2015}$ after division with $x^2+x+1$

Remainder of $(1+x)^{2015}$ after division with $x^2+x+1$ Is it correct if I consider the polynomial modulo $5$ $$(1+x)^{2015}=\sum\binom{2015}{n}x^n=1+2015x+2015\cdot1007x^2+\cdots+x^{2015}$$ ...
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3answers
120 views

Showing that $x^4 -2x^2 +8 x+1$ is irreducible over $\Bbb Q$

I want to show that the polynomial $$f(x)= x^4 -2x^2 +8 x+1$$ is irreducible over $\Bbb Q$. I've proved it by a long method, but I need an easy and short method. I've try to put $x=t+1$, but this ...
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2answers
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How come number if reducible polynomials becomes larger than total number of polynomials over $\mathbb Z_3$?

I was searching for the numeber of monic irreducible polynomial of degree 3 in $\mathbb Z_3[x]$ (please note that I have found this problem in Contemporary Abstract Algebra by Gallian and hence am ...
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1answer
122 views

Radical solution to a polynomial quartic equation

Consider the following quartic equation: $$x^4 + rx^3 + r^2x^2 + r^3x + r^4 - 1 = 0$$ By Lodovico Ferrari solution, this equation must possess four radical solution provided that $r$ is a rational ...
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1answer
45 views

Derivation of Viète's Theorem or Formulas

I had problems finding the proof of an equation in G. Polya "Mathematics and Plausible Reasoning" p. 18 that upon a little bit of research turns out to be Viète's Theorem: Given a polynomial, $$a_0 ...
3
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3answers
49 views

Zeroes of polynomials and their sum

Let $a, b$ are zeroes of the polynomial $x^2-10cx-11d$ and $c,d$ are the zeroes of the polynomial $x^2-10a x-11b $ where $a,b,c, d$ are distinct reals then $a+b+c+d=?$
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2answers
93 views

Showing that $x^3+2y^3+4z^3=2xyz$ has no integer solutions except $(0,0,0)$.

Let $x,y,z\in \mathbb{Z}$ satisfy the equation: $$ x^3+2y^3+4z^3=2xyz $$ How do I prove that the only solution is $x=y=z=0$?
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2answers
53 views

Curve Fitting to Represent Any Data

I'm a programmer seeking to take a bunch of data and represent it as a curve. Specifically, I want to take several hundred/thousand (floating) points and represent those points to a specified level of ...
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0answers
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Roots of a polynomial

I am working the next problem: Consider the polynomials $$ p_n(z)=\sum_{j=0}^{n}\frac{z^j}{j!} $$ For $n \geq 2$, show that if $a \in \mathbb{C}$ is such that $|a|=1$ or $|a|=n$, then ...
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4answers
54 views

Proof of Different Polynomial Decompositions into Linear Factors

From G. Polya "Mathematics and Plausible Reasoning" p. 18. How do you prove that provided the roots of a polynomial are different from zero, $$a_0 + a_1x+a_2x^2 + ... + a_nx^n$$ $$\,= ...
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1answer
27 views

Common factor in series solutions

a few times during my lectures on series solutions ( to ODE) the teacher mentioned that it was only valid to use all the theorems and methods and such on analytic polynomials if they did not have a ...
5
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1answer
44 views

A question about a polynomial

Suppose that $p$ is a real polynomial of degree $n$. Prove that for $|x|<1$, $$\sum\limits_{m=0}^\infty{p(m)x^m}=h((1-x)^{-1})$$ for some real polynomial $h$ of degree $n+1$ without the ...
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2answers
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Maximum number of roots/zeros a polynomial $ax^4+bx+c$ can have?

I need to find the maximum number of roots the polynomial $g(x)=ax^4+bx+c$ defined on interval $[-1,1]$ can have. Intuitive and geometrical example shows that it can have no more than two. My method ...
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0answers
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Is my proof showing that if products of powers of elementary symmetric polynomials are the same, then the power is same, correct?

Below proof is assuming this lemma, which can be proven easily: Let $\leq$ be a monomial ordering on $R[X_1,...,X_n]$. Let $f,g\in R[X_1,...,X_n]$ be nonzero polynomials such that $LC(f)$ is ...
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4answers
56 views

Proof that a polynomial has a minimum in $\Bbb R$

I have to prove to following statement and I am having a really hard time here. There it is: Prove that the following polynomial has a minimum in $\Bbb R$ $$p(x)=x^4 + a_3x^3 + a_2x^2 + a_1x + ...
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2answers
37 views

Show that every $f(x) \in K[x]$ can be represented as $g(x^{p^e})$

A (probably simple) question I encountered but I don't know how to approach: Let $K$ be a field of prime characteristic $p>0$. Show every $f(x) \in K[x]$ can be represented as $g(x^{p^e})$ ...
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0answers
19 views

Old and recent results concerning the number of real roots for a polynomial function

Let $$p(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots +a_0$$ be a real polynomial function with real coefficients. My question is: I want to make a list of old and recent results concerning the number of real ...
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2answers
52 views

The product of two of the four roots of $x^4 -20x^3+ kx^2 + 590x -1992 = 0$ is $24$ the find $k$. [closed]

Please help. I tried to solve by taking sum of roots as $20$ and product as $1992$. No idea how to proceed. Thanks in advance
8
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1answer
131 views

Proving that a polynomial of the form $(x-a_1)\cdots(x-a_n) + 1$ is irreducible over $\mathbb{Q}$

I want to prove that for any set of distinct integers $a_1,\ldots,a_n$, the polynomial $$h = (x-a_1)\cdots(x-a_n) + 1$$ is irreducible over the field $\mathbb{Q}$, except for the following special ...
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0answers
32 views

What will happen if there is a way predicting at a least one root of $p_{n}(x)=0$ without calculator?

let $p_{n}(x)$ be a polynomial of degree $n$ defined as follow : $p_{n}(x)=x^n +a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+.....a_{0}$ which : $a_{n-1},a_{n-2},.....,a_{0}$ are non nul real numbers coefficients. ...
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1answer
33 views

How to find one perpendicular to the basis of the set? [closed]

In the set of real polynomials, consider the inner product given by $$\langle p,q\rangle = \int_0^1 p(x)q(x)dx$$ How do I find a polynomial perpendicular to both elements of the set $\{1 + t, t^2 - ...
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1answer
28 views

Quadratic Diophantine equations on the ring of polynomials

The set of solutions of quadratic equation $a^2+b^2=c^2$ on $\mathbb{Z}$ can be described by Pythagorean triples up to multiplication. Can I use similar results on the ring of integer coefficient ...
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2answers
56 views

Finding $n$th root of 2 is irrational using given polynomial

The polynomial $f(x)$ is defined by $f(x)=x^n + a_{n-1}x^{n-1}+ \cdots + a_{2}x^2+a_1x+a_0$ where $n \geq 2$ and the coefficients $a_0, \cdots, a_{n-1}$ are integers, with $a_0 \neq 0$. ...
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1answer
26 views

Why does it mean that $n$-th variable is removable?

I'm reading the proof for "the fundamental theorem of symmetric polynomials" and I have a trouble with it (http://en.m.wikipedia.org/wiki/Elementary_symmetric_polynomial) Let $P(X_1,...,X_n)$ be a ...
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1answer
14 views

For any monomial ordering, $1\leq m$ for any monomial $m$

Let $R$ be a ring. Let $\leq$ be a well-ordering on the set of (monic) monomials in $R[X_1,...,X_n]$. Then, $\leq$ is said to be a monomial ordering iff $mm_1\leq mm_2$ whenever $m_1\leq ...
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3answers
35 views

Method for proving polynomial inequalities

Let $x\in\mathbb{R}$. Prove that $\text{(a) }x^{10}-x^7+x^4-x^2+1>0\\ \text{(b) }x^4-x^2-3x+5>0$ Possibly it can be proved in a few different ways, but I have first tried to prove it ...
3
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2answers
28 views

irreducibility of polynomials made by perturbation from a polynomial

Suppose $f(x)\in\mathbb{Z}[x]$ with $\text{deg}f=2n,n\in\mathbb{Z_+}$ and $f_m(x):=f(x)+ mx^n $ for each integer $m\in\mathbb{Z}$. Let us define a number $P_f$: ...
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1answer
23 views

Maximum modulus of a holomorphic function on a disc within a certain sector

Given the polynomial $$f(z) = az^n + b \qquad (n \geq 2)$$ and a modulus $0 < \rho < 1$, can one find a modulus $0 < r < \rho$ such that there is a point $$w \in \{ |z| \leq r \} \cap \{ ...
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1answer
15 views

Monomial ideals and Dickson's lemma

I am currently revising for my exams and working on questions about monomial ideals and came across this question. Let I be the ideal of $\mathbb{R}[x,y]$ generated by all polynomials of the form ...
5
votes
4answers
65 views

Prove that $f=x^6+ax+5$ is reducible over $\mathbb{Z_7},\forall a\in\mathbb{Z_7}$

We have $f=x^6+ax+5\in\mathbb{Z_7}$ and we have to show that it is reducible on $\mathbb{Z_7}$, $\forall a\in\mathbb{Z_7}$. Here are all my steps: For $a=0$ we'll get $f=x^6+5\in\mathbb{Z_7}$. But ...
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1answer
50 views

Fraction modulo integer in sage [closed]

I'm working on a sage script right now, I have some polynomials coefficients that are rational, and I want to apply a congruence on these coefficientss, for example: $p = 1 + (7/2)x$ the function ...
0
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1answer
40 views

Are Zero Degree polynomials Considered monics?

DO zero degree polynomials that is constant polynomials considered monic polynomials? Example F(x)=16 Does it Matter the Field or the Integral region where i take the coeficients from?Sorry if the ...
0
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1answer
55 views

How is the degree of a polynomial defined? $a_1+a_2x^2+\cdots+a_nx^{n-1}$ has degree $n$ or $n-1$?

I have this polynomial: $$a_1+a_2x^2+\cdots+a_nx^{n-1}$$ or: $$a_0+a_1x^2+\cdots+a_{n-1}x^{n-1}$$ What is degree of those polynomials? $n$ or $n-1$, I'm little bit confuse... Thank you!