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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Do perfect square trinomials only have one root?

I apologize for the basic question, but I'm just now learning of perfect square trinomials in my math class. Google hasn't provided any relevant answers. Throughout all of the examples I have been ...
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5answers
44 views

Find the cubic equation of roots $α, β, γ$.

Taken from Fitzpatrick $4$ unit course textbook. The question says: If the cubic equation $\ ax^3+bx^2+cx+d$ has roots $α, β, γ$. Find the cubic equation who's roots are $α^2, β^2, γ^2$ I keep ...
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3answers
796 views

Is there an algorithm to compute the degree of a polynomial?

Let $f\in k[X]$ be a polynomial in one unknown over any field (or any nice enough commutative ring, I imagine - it shouldn't matter) and suppose that all we can do to understand $f$ is to evaluate it ...
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1answer
16 views

Problem regarding polynomials and partial derivatives

Let $P:\mathbb{R}^n\rightarrow\mathbb{R}$ be the homogeneous polynomial of degree $k$: $$P(x)=\sum_{|a|=k}c_{\alpha}x^{\alpha}$$ How can I show: $\partial^{\beta}P(x)=\beta !c_{\beta}$ for all ...
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1answer
25 views

Find the parameter $m$ such that the number always be perfect square.

I have to find the values of the parameter $m\in\mathbb{Z}$ such that the polynomial $(x-1)(x+3)(x-4)(x-8)+m$ is always a square for any $x\in\mathbb{Z}$. Any hint?
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0answers
10 views

Radical extension and algebraic solution of an irreducible polynomial

Suppose that $k$ is a field with characteristic equal to zero, that $P \in k[X]$ is an irreducible polynomial and that $\alpha$ is a root of $P$ in an algebraic closure $\overline{k}$. Suppose also ...
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1answer
33 views

Every irreducible polynomial f over perfect field F is separable

Every irreducible polynomial f over perfect field F is separable. Can you check my proof? Let f is inseparable. So we have $f=\sum_i h_ix^i$ and $f^p=\sum_i h_i^px^{ip}$ Now I use Frobenius mapping ...
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5answers
551 views

Systematically guessing integer roots of a cubic polynomial

Suppose I have a cubic equation, such as $$15x^3-4x^2-25x+14=0.$$ By the Hit and Trial method I know that one of the roots is $x=1,$ and hence I can solve the cubic equation with ease, as it will ...
11
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1answer
418 views

Sum of roots of cubic = -coefficient of quadratic term?

Working through Ian Stewart's "Galois Theory, Third Edition," he states at the end of the second paragraph on page 13: "Because we know that $\alpha_1+\alpha_2+\alpha_3$ is minus the coefficient of ...
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2answers
35 views

Role of known term in Routh - Hurwitz criterion, for $x^8 - 36·x^7 + 546·x^6 - 4536·x^5 + 22449·x^4 - 67284·x^3 + 118124·x^2 - 109584·x + 40321=0$.

I was studying the sign of the solutions of this polynomial. $$x^8 - 36·x^7 + 546·x^6 - 4536·x^5 + 22449·x^4 - 67284·x^3 + 118124·x^2 - 109584·x + 40321=0$$ I tried to apply the Routh-Hurwitz ...
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0answers
45 views

How to do polynomial composition/substitution? (Vincent-Alesina-Galuzzi)

For the polynomial $$ p(x) = \sum_{i=0}^n c_i x^i, $$ of real coefficients and real variable, obtain the coefficient of $$ q(x) = \left(1 + x\right)^n p\left( \frac{a + b x}{1 + x} \right), $$ as ...
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2answers
227 views

Proof that every point can lie on a tangent to a curve

for which odd degree polynomials will every point in the plane lie on at least one tangent to the curve p(x)? What if P(x) is even?
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0answers
15 views

Multivariable polynomial matrix representations

This is a follow-up to matrix representation of parabola and matrix representation to generate monomials. I found a method to build such matrices to implement this type of functionality for one ...
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1answer
15 views

Codimension of the image of the polynomials subspace is infinite

Consider the interval $I=[0,1]$ and the Banach space $E$ of real continuous functions defined on $I$ ($E=\mathcal C_{\mathbb R}(I))$. $P \subset E$ is the subspace of polynomial functions (restricted ...
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0answers
23 views

Unknown variable in formula - binomial coefficient? [duplicate]

I'm currently researching a filter and don't quite understand one of the equations used there, since it contains a variable I don't know how to calculate: $$(1)\,\,a^{m, k}_s = \frac{c^{k, ...
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1answer
16 views

Deriving uniform cubic B-spline matrix

The page 6 of this paper says: Condition 1: $p(0) = q(1)$ – Symmetry: $p(0) = q(1) = 1/6(\pi-2 + 4 \pi-1 + \pi)$ Condition 2: $p’(0) = q’(1)$ – Geometry: $p’(0) = q’(1) = 1/2 ((\pi – \pi-1) + ...
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1answer
39 views

Roots of $p(x)=\prod_{i=1}^{2n}(x-d_i)+k^2, \ \ \ \ n\in\mathbb N,\ k\in\mathbb R$

Let $$p(x)=\prod_{i=1}^{2n}(x-d_i)+k^2, \ \ \ \ n\in\mathbb N,\ k\in\mathbb R$$ where $d_i>0$ for all $i=1,\dots,2n$. Can I infer that $$p(x)=0$$ has only roots with positive real part?
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2answers
51 views

A question about the coefficient of a specific polynomial

Let $$ \begin{aligned} f &= f(x_1,x_2,x_3,y_1,y_2,y_3,z_1,z_2,z_3)\\ &=(x_2+x_3-y_1-z_1)(x_1+x_3-y_2-z_2)(x_1+x_2-y_3-z_3)\\ &\times(y_2+y_3-x_1-z_1)(y_1+y_3-x_2-z_2)(y_1+y_2-x_3-z_3)\\ ...
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1answer
24 views

Polynomial inequalities of the form $C P_2 \leq P_1 \leq D P_2$

Let $P_1$ and $P_2$ be polynomials in $\mathbb{R} [x_1, \ldots, x_n]$ of the same degree. Under what conditions are there $C,D \in \mathbb{R}$ so that $C P_2 \leq P_1 \leq D P_2$ (as functions)? ...
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51 views

Factor a cubic polynomial

Is there a simple way to find out that, for example, $u^3 - 54u + 108$ is $(u - 6)(u^2 + 6u - 18)$?
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1answer
27 views

Looking to understand proposition related to the fundamental theorem of algebra

I am having some problem understanding exactly what the following proposition is saying. Also, is this result have a common name? How important it is, etc. It is $\mathbf{Proposition:}$ Let ...
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2answers
63 views

Six variables. System of equations.

$$ \begin{align} x & =\frac{R+\frac{G+B}{-2}}{R+G+B} \\[10pt] y & =\frac{\frac{(G-B) \sqrt{3}}{2}}{R+G+B} \\[10pt] z & =R+G+B \end{align} $$ How do I get the formula for ...
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0answers
34 views

How to solve this kind of recurrence relation in closed form? $F(n) = aF(n-1) + bF(n-2) + cF(n-3) + dF(n-4)$

How to solve this recurrence relation in closed form? $$F(n) = aF(n-1) + bF(n-2) + cF(n-3) + dF(n-4)$$ I know how to solve recurrence relations for less than four calls by solving the ...
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1answer
20 views

polynomial over a field, applied onto a Jordan block

Let $K$ be a field of characteristic $0$, $f \in K[t]$ a polynomial over $K$ and $J \in M_{n,n}(K)$ a Jordan block to an eigenvalue $\lambda \in K$, meaning that $J$ has the shape: $$J = ...
0
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2answers
57 views

Common roots of degree 4 polynomials [on hold]

Find the common roots of $x^4+5x^3-22x^2-50x+132=0$ and $x^4+x^3-20x^2+16x+24=0$ Hence solve the equation
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2answers
38 views

Kolmogorov-Zurbenko filter - Calculation of coefficients

I'm currently researching the Kolmogorov-Zurbenko filter and trying to implement it myself as a way to smooth one-dimensional signal strength values. The basic filter per se is pretty easy to ...
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3answers
180 views

matrix representations and polynomials

I just investigated the following matrix and some of its lower powers: $$M = \left[\begin{array}{cccc} 1&0&0&0\\ 1&1&0&0\\ 1&1&1&0\\ 1&1&1&1 ...
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1answer
22 views

polynomial modulo for higher degree

Given $f(x) , n, g(x)$ where $g(x)$ is usually of a small degree then if we find $h_1(x)$ such that $f(x)\equiv h_1(x)\mod \{n,g(x)\}$ , Is there any algorithm to find $h_2(x)$ such that $f(x)\equiv ...
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2answers
31 views

Positive roots of polynomial $q(x)=p(x)+k^2$

Let $p(x)$ a polynomial of degree $n\in\mathbb N$ such that $$p(x)=0$$ has exactly $n$ real and positive solutions. Is it true that polynomial $q(x)=p(x)+k^2$, for $k\in\mathbb R$ has only positive ...
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2answers
46 views

What is the general definition of a discriminant? (Not just the definition for polynomials)

For example, in regards to the second derivative test for a function of two variables, $D=f_{xx}f_{yy}-(f_{xy})^2$ is refered to as the "second derivative test discriminant." I know that D is the ...
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1answer
50 views

Prove the function is nondecreasing

Lets take: $A_1,...,A_n$ family of finite, nonempty sets. Define: $$f(t)=\sum_{k=1}^n\left( \sum_{1\le i_1<...<i_k\le n}(-1)^{k-1}t^{|A_{i_1} \cup ... \cup A_{i_k}|} \right)$$ for $t \in [0,1]$. ...
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1answer
13 views

Bijections Between Different Branches of the Inverse of a Polynomial of a Single Complex Variable

Let $F\left(z\right)$ be a polynomial of degree $d≥2$ with complex coefficients. Let $D$ be an open disk in the complex plane containing no critical points of $F\left(z\right)$. Let ...
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28 views

How to transform one polynomial curve into another

I have a problem: I have two polynomial splines: For red curve I have used Rational fraction function with p=3/q=2 and for blue curve I have also used rational fraction function with p=1/q=3 It ...
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8 views

Duality gap analysis

I solved a non-linear non-convex optimization problem via dual decomposition optimization using sub-gradient method. (my main goal is to solve the problem in a distributed way). I solve the same ...
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1answer
77 views
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Interpolation of polynomial.

Suppose we have the polynomial $f(x)=x^3$. We can now interpolate it using the values: $$f(1)=1,f(2)=8,f(3)=27,...$$ Using only one value, we get a constant: $$f_1(x)=1,\;\{1,1,1,...\}$$ Now using two ...
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1answer
52 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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1answer
32 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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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
26 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
70 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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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
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 + ...
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1answer
40 views

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 ...
6
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1answer
47 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
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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 ...
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1answer
16 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 ...
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3answers
67 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 $z^3 + 3z^4 + 6z^5 + 10z^ 6 + 12z^ 7 + 12z^ 8 + 10z^ 9 + 6z^ {10} + 3z^ {11} + z^{12}$. ...
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54 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) [closed]

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)
3
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1answer
54 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 ...
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 ...