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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2
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1answer
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Roots of polynomial outside a vertical strip of $\mathbb C$

Let $P(z)$ be an arbitrary polynomial with real coefficients. I'd like to guarantee that all roots of $P$ have real parts outside the interval $(0, 1)$. Is there some simple condition on P that will ...
-2
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0answers
19 views

Uncommon Rational Function Expansion [on hold]

I am totally surprised by this awesome expansion: $$ \frac{a_0 + a_1x + a_2x^2 + a_3x^3}{b_0 + b_1x + b_2x^2} =\\ -\frac{a_3 b_1 - a_2 b_2}{b_2^2} +\frac{a_3x}{b_2} +\frac{a_3 b_0 b_1 ...
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0answers
12 views

Matrix representations of parabola.

Continuing the epic quest on finding matrix representations from here: Representation of hyperbolas. with a last part, the only conic section left: the parabola. I will present one idea of how to ...
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0answers
14 views

Quadratic polynomials describe the diagonal lines in the Ulam-Spiral

I'm trying to understand why is it possible to describe every diagonal line in the Ulam-Spiral with an quadratic polynomial $$2n\cdot(2n+b)+a = 4n^2 + 2nb +a$$ for $a, b \in \mathbb{N}$ and $n \in ...
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0answers
5 views

How to build an 2-D polynomial from 1-D orthogonal polynomials

I have an set of orthogonal polynomials such as I want to build an 2D polynomial following the equation $$P_k(x,y)=P_k(x)P_k(y)$$ where $k=1..4, (x,y) \in [-1, 1]^2$ Based on given $P_n(x)$ as ...
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1answer
22 views

Which polynomial has similar properties with Legendre?

I am looking for an kind of polynomial such as Legendre properties that polynomial sequence of orthogonal polynomials such as bellow image. Could you suggest to me one polynomial? Is B-spline correct? ...
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2answers
51 views

Find conditions for $a$ and $b$ such that $P(x)=x^4-(a+b)x^3+(ab+2)x^2-(a+b)x+1$ has only real roots.

I need to find conditions for a and b such that $$P(x)=x^4-(a+b)x^3+(ab+2)x^2-(a+b)x+1$$ has only real roots. Any hints on how I should do that?
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2answers
280 views

Polynomial rings — Inherited properties from coefficient ring

To avoid mixing up things, I wanted to collect properties a polynomial ring is inheriting from the coefficient ring and what property implies another. Let $R$ be a ring (what else do I need at which ...
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0answers
33 views

seeking sufficient conditions for polynomials to have no positive roots

I encountered several polynomials as below: $$f(x)=7 + 91 x - 385 x^2 + 1659 x^3 - 1379 x^4 + 553 x^5 - 35 x^6 + x^7$$ $$g(x)=33 + 110 x + 495 x^2 - 252 x^3 + 335 x^4 - 18 x^5 + x^6$$ $$h(x)=71 + ...
3
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1answer
51 views

Polynomial-closed properties of rings [on hold]

If $R$ is a ring with certain property, sometimes when we pass to the polynomial ring in one variable, the ring $R[x]$ still has the same property. For instance, it's a theorem that if $R$ is a UFD ...
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1answer
115 views

Extremely difficult: Polynomials $f,g,h$ such that…

I've been trying to solve this for hours and got nowhere, so I can only assume it's a really difficult problem. Problem: Find polynomials $f,g,h$ with integer coefficients such that: ...
2
votes
1answer
327 views

The volume in cubic meters of water in an aquarium…

The volume in cubic meters of water in an aquarium is given by the polynomial $$V(x) = x^3 - 16x^2 + 79x - 120\;.$$ If the depth in feet (it really does say feet) can be represented by $x-3$, what ...
6
votes
1answer
41 views

Cyclotomic polynomials, properties.

Let $F$ be a field of characteristic prime to $n$, and let $F^a$ be an algebraic closure of $F$. Let $\zeta$ be a primitive $n$th root of unity in $F^a$. I know that the monic polynomial $\Phi_n(X)$ ...
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0answers
5 views

Question on real polynomial in projective space

Hi all I was given this question and desperately in need of help. I am given a homogeneous polynomial of degree 4 of two variables x and y, with real coefficients with 4 real distinct projective roots ...
2
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3answers
66 views

How to solve $(z^1+z^2+z^3+z^4)^3$ using Pascals Triangle?

In an exercise it seems I must use Pascal's triangle to solve this $(z^1+z^2+z^3+z^4)^3$. The result would be $z^3 + 3z^4 + 6z^5 + 10z^ 6 + 12z^ 7 + 12z^ 8 + 10z^ 9 + 6z^ {10} + 3z^ {11} + z^{12}$. ...
1
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1answer
15 views

Constructing matrices with eigenvalues equal to roots of a given polynomial

Suppose we are given a polynomial e.g. $$x^4+Ax^3+Bx^2+Cx+D,\tag1$$ and we need to construct a matrix, whose eigenvalues would be equal to roots of this polynomial. One way, rather inelegant, is to ...
2
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2answers
682 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
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0answers
39 views

Please I would like to have the solution of the exercise 3.7 on page 105 of The Arithmetic of elliptic curves second edition (J. H. Silverman) [on hold]

Please I would like to have the solution of the exercise 3.7 on page 105 of The Arithmetic of elliptic curve second edition (J. H. Silverman)
5
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1answer
64 views

Function equation, find the function evaluated at the certain point.

Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$, $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$ The constant term, $a_0 = f(0) = 1$. Let: ...
5
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1answer
159 views

$15a+6b+4c+8d=0$ implies $ax^3+bx^2+cx+d$ has a positive root

Let $a,b,c,d$ be real numbers such that $15a+6b+4c+8d=0$. Show that $f(x)=ax^3+bx^2+cx+d$ has a positive root. (Komal, Problem N. 170.) I want to try to use the intermediate value theorem, showing ...
3
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1answer
20 views

finding polynomials to approximate a multivariable function

Let $U := B_1(0) \subseteq \mathbb{R}^2$, with $B_1(0) := \{(x, y) \in \mathbb{R}^2,\space \|(x, y)\| _1 < 1\}$. Now consider the function: $$g: U \to \mathbb{R}^2, (x, y) \mapsto ...
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1answer
172 views

Ladder against a wall. [duplicate]

Having a bit of a problem with a question. There is a 4m ladder leaving against a wall. There is a box in between The ladder and wall. The box is a cubic metre. I have found a quartic to find the ...
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3answers
1k views

How to factorize $x^3 - 7x + 6$?

How do you factorize this polynomial: $${x^3 - 7x + 6}$$ Some online solver doesn't even work saying: using GCF method doesn't work, but sites like Mathway.com gave me the answer, is there a ...
2
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1answer
34 views

Polynomial division simple

I am trying to divide polynomials but i am ending up with different outcomes. Let's assume i have : $\dfrac{s^4+3s^3+4s^2+4s+1}{2s^3+2s^2+3s+1}$ Can anyone solve this step by step (long or synthetic ...
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1answer
31 views

Any shortcut method to compare the roots of two quadratic equations? [on hold]

The given equations are(for example) $81x^2-9x-2=0$ and $56y^2-13y-3=0$. How do i compare the roots of these equation without using the Quadratic formula? Any suggestions please? Thanks.
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3answers
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Are polynomials infinitely many times differentiable?

Are polynomials infinitely many times differentiable? If so, does it only mean that at some point we reach 0 and then we keep on getting 0? Thank you!
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3answers
585 views

Why are the coefficients always equal?

Take the equation $ax^{2} + bx + c = 3x^{2} + 4x + 53$. Why is it always true that $a = 3, b = 4$ and $c = 53$? I've seen many examples like this where the coefficients are equated, and was just ...
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3answers
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How can I prove irreducibility of polynomial over a finite field?

I want to prove what $x^{10} +x^3+1$ is irreducible over a field $\mathbb F_{2}$ and $x^5$ + $x^4 +x^3 + x^2 +x -1$ is reducible over $\mathbb F_{3}$. As far as I know Eisenstein criteria won't ...
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0answers
11 views

Measure of variation(?) of multidimensional polynomial function

I have a multidimensional function $$\mathbf{f}(x) = [f_0(x), ... , f_N(x)]$$ where $f_n$ are real-valued trigonometric polynomials. I want to measure how much $\mathbf{f}(x)$ varies over some ...
3
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2answers
58 views

Shamir's secret sharing interpolation problem

I try to understand this protocol - Shamir's secret sharing - threshold scheme. I got my data and I made interpolation basing on examples published on Wikipedia. You can see them below (sorry, I am ...
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3answers
133 views

Why can't a non-zero polynomial satisfy some equations?

I'm having a hard time visually picturing/understanding how to explain why a non-zero polynomial function cannot satisfy the equation: $f''(x)$ = $-f(x)$ So is it basically asking to explain why a ...
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2answers
303 views

Graeffe's root finding method

What are the practical applications of Graeffe's root finding method?I searched a lot but couldn't find.I found that it is used in aerodynamics and electric circuit analysis.But don't know much about ...
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3answers
27 views

Constructing Polynomial Function from Set of Points and Slopes

I only have a basic knowledge of calculus but I would like to know if it's possible to, given a set of points each with their own slopes, construct the simplest (or any) polynomial function that ...
0
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3answers
66 views

Polynomial whose one of its roots is $\cos(\pi/7)$

Let $P(x)$ be a 3rd-degree polynomial with integer coefficients, one of whose roots is $\cos(\pi/7)$. Compute $\frac{P(1)}{P(-1)}$ I saw this question in a contest math problem, and I know that it ...
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4answers
69 views

Let $f(x)$ be polynomial of degree four [on hold]

Let $f(x)$ polynomial of degree four where: $$f(1)=1,f(2)=4,f(3)=9,f(4)=16, f(7)=409$$ Find $$f(5)=??$$
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4answers
76 views

Suppose that $\alpha$ root of the equation [on hold]

Suppose that $\alpha$ root of this equation: $$x^4+x^2-1=0$$ Find the value of $$\alpha ^{6}+2\alpha ^{4}$$ "I want the way, not the roots of the equation." I tried, but I couldn't find any thing.
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2answers
32 views

Three polynomials as unknowns of an equation

If three polynomials $f,g,h\in\mathbb R[x]$ are such that $[f(x)]^2 –x[g(x)]^2+[h(x)]^2=0$, what can we conclude about $f, g, h$?
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0answers
9 views

Trace of an element in a separable field extension

Let $L=K(\alpha)$ be a finite separable field extension of $K$ of degree $n$ and let $\alpha$ have minimal polynomial $f(X)\in K[X]$ with roots $\alpha=\alpha_1,...,\alpha_n$. Write ...
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1answer
28 views

Example of $Q((x))$ that doesnt match field of fractions of ring $F[[x]]$

Let $F$ be a commutative ring without zero divisors and $Q$ -its field of fractions. Let $Q(x)$ be also field of fractions of ring $F[x]$. How can field $Q((x))$ not match field of fractions of ring ...
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1answer
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All polynomials with no natural roots and integer coefficients such that $\phi(n)|\phi(P(n))$

Let $P(x)$ be a polynomial with integer coefficients such that the equation $P(x)=0$ has no positive integer solutions. Find all polynomials $P(x)$ such that for all positive integers $n$ we have ...
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2answers
36 views

Find the sum of the roots of the exponential equation

The equation $$2^{333x - 2} + 2^{111x + 2} = 2^{222x + 1} + 1$$ has three real roots. Find their sum. I'll simplify it first as: $$\frac{1}{4}2^{333x} + (4)2^{111x} = (2)2^{222x } + 1$$ Let ...
2
votes
3answers
52 views

Find the sum of the roots given no multiple roots.

Find the sum of the roots, real and non-real, of the equation $$ x^{2001} + \left( \frac{1}{2} - x \right)^{2001} = 0 $$ given that there are no multiple roots. I am in a weird situation here. ...
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2answers
40 views

Proof of associativity of polynomials product (infinite variables)

The product of polynomials in $R[X_i]_{i\in I}$ where $I$ is not necessarily finite is associative ($R$ commutative ring), but I can't find any detailed proof of this fact. Either it is left in ...
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1answer
390 views

Algorithm to find the maximum/minimum of a polynomial without graphing.

For a quadratic equation of the form $y=ax^2+bx+c$ the max/min occurs at $x=-\frac{b}{2a}$. Is there any hard and fast equation like this for polynomials of degree $\geq 4$?. For such polynomials the ...
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3answers
40 views

Multiplicity of a root of a polynomial

:) It's true that, if a polynomial has a root (let's say, k, for example) with multiplicity n (n>1, for n integer), then it's true that the derivate polynomial have k as a root with multiplicity ...
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1answer
71 views

A polynomial that satisfies $x^pf(1-x) + (1-x)^pf(x) = 1$

The context of this question is the construction of the Daubechies wavelet. $f$ is a polynomial of degree $p-1$ which satisfies the equation: $$ x^pf(1-x) + (1-x)^pf(x) = 1 \tag{1} $$ Since $$ ...
2
votes
1answer
249 views

Let $f:R\longrightarrow S$ be a surjective ring homomorphism. If $R$ is PID, then $S$ is PIR.

Let $f:R\longrightarrow S$ be a surjective ring homomorphism. If $R$ is PID, then $S$ is PIR. I think I have proved this: Let $J$ be an ideal of $S$. Then $f^{-1}(J)=(a)$ is a principal ideal of ...
0
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1answer
13 views

module isomorphism inbetween two equivalence classes of polynomials

Let $g \in \mathbb{R}[t]$ be a normed irreducible polynomial of degree 2, meaning that $g(t) = (t - \lambda)(t - \overline{\lambda}$) for a $\lambda = a + b i$, with $a, b \in \mathbb{R}$, $b ≠ 0$. I ...
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2answers
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6
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81 views

Proving that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$.

I need to prove, that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$. Prove that $$x^m+x^{-m}=P_m (x+x^{-1} )=a_m (x+x^{-1} )^m+a_{m-1} (x+x^{-1} )^{m-1}+...+a_1 (x+x^{-1} )+a_0$$ on ...