Questions tagged [polynomials]

Questions on the functionality operating on polynomials

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49
votes
4answers
16k views

How do I work with Root objects?

I want to solve the trigonometric equation : $$(3-\cos 4x )\cdot (\sin x - \cos x ) = 2.$$ I tried Solve[(3 - Cos[4*x])*(Sin[x] - Cos[x]) == 2, x] It returns the ...
50
votes
6answers
11k views

Finding real roots of negative numbers (for example, $\sqrt[3]{-8}$)

Say I want to quickly calculate $\sqrt[3]{-8}$, to which the most obvious solution is $-2$. When I input $\sqrt[3]{-8}$ or Power[-8, 3^-1], Mathematica gives the ...
48
votes
3answers
3k views

Funny behaviour when plotting a polynomial of high degree and large coefficients

I am trying to plot a polynomial of degree $29$ on the domain $[0,1]$, with fairly large coefficients: ...
32
votes
5answers
3k views

How do I reassign canonical ordering of symbols?

I have a big polynomial that evaluates to: $$A^2 e^2 \phi ^- \phi ^++A e \phi ^- \phi ^+ c_{2 w} g_Z+\frac{1}{2} A e g h W^- \phi ^+ +\ll13\gg,$$ which is supposed to represent some terms in the ...
20
votes
4answers
3k views

How to keep Collect[] result in order?

For example, Collect[(1 + x + Cos[s] x^2)^3, x] gives the result ...
11
votes
3answers
8k views

First positive root

Simple question but problem with NSolve. I need help how to extract first positive root? For example: ...
21
votes
7answers
14k views

How do I replace a variable in a polynomial?

How do I substitue z^2->x in the following polynomial z^4+z^2+4? z^4+z^2+4 /. z^2->x ...
18
votes
4answers
16k views

Factoring polynomials to factors involving complex coefficients

I've run into some problems using Factor on polynomials with complex coefficient factors. Reading the documentation it looks like it only factors over the ...
13
votes
3answers
10k views

Get polynomial interpolation formula

I'm attempting to get a polynomial interpolation formula out of Mathematica but I am absolutely lost. I stared out using ...
7
votes
2answers
1k views

Why does Expand not work within a function?

I'm writing this fairly simple function: ...
5
votes
3answers
1k views

How can I reorder the factors in the terms of a polynomial?

How can I reorder the factors in the terms of a polynomial? Consider ...
13
votes
2answers
5k views

Gram-Schmidt Process for Polynomials

I want to implement the Gram Schimdt procedure to the vector space of polynomials of degree up to 5, i.e. I want to find an orthogonal basis from the set of vectors $v=(1,x,x^2,x^3,x^4,x^5)$. The ...
17
votes
5answers
1k views

Decomposing a matrix with polynomial elements into a polynomial with matrix coefficients

Let's say I have a matrix, $\mathbf{M}$, that is polynomially dependent on a single variable, such as M = {{15 + a^2, a + 5 a^2}, {a - 5 a^2, 2}} and I want to ...
11
votes
3answers
3k views

How can we plot the complex roots of an equation?

If we'd like to display the $n$ roots of a polynomial on the complex plane as points, how can we do this? For example, if we have the equation $x^3 + x^2 + x + 1$, how can we plot the 3 roots as ...
10
votes
3answers
1k views

How do I get a two-term polynomial with a leading negative sign to display in the correct (i.e. textbook) order?

The first three expressions evaluate as expected and the polynomial is displayed in what I would call "textbook" form. The last expression, however, switches the order of terms. Mathematica employs ...
14
votes
2answers
1k views

Factorizing polynomials over fields other than $\mathbb{C}$

I'd like to take a polynomial in $\mathbb{Z}_5[x]$ of the form $ax^2+bx+c$ and factor it into irreducible polynomials. For example: Input... x^2+4 Output... <...
9
votes
3answers
2k views

Checking if the roots of a function are real

I'm trying to determine if the roots of a function are real. How would you do that? (In particular I'm interested in verifying that the roots of LegendreP[6, x] ...
16
votes
2answers
605 views

How can I get the solutions of $x^n-x-t=0$ in hypergeometric form?

$x^3-x-t=0$ has three roots that can be expressed in hypergeometric form, for example $$x_1=-1-\frac{t}{2}{_2F_1\left(\frac{1}{3},\frac{2}{3};\frac{3}{2};\frac{27t^2}{4} \right)}+\frac{3t^2}{8} {...
6
votes
5answers
2k views

Convert polynomial to Chebyshev

I want to convert a polynomial in "standard form" to Chebyshev form. Here's one way to do it: ...
4
votes
1answer
874 views

Small Issue with Chebyshev Derivative Approximation

I am trying to approximate the derivative of a function from its Chebyshev expansion. I start out with the following random function: ...
18
votes
2answers
5k views

Why does Mathematica order polynomial forms in reverse from traditional order?

I could very well be missing something obvious, but this has always bugged me with Mathematica and I don't know why it does it or how to fix it. If I enter any polynomial, say, x^2 + x - 1 for ...
7
votes
1answer
3k views

How to substitute the following conditions into an expression?

I have an expression: $p=a\;b\; x + b^2\; y + a\;c\; z$. I want to substitute $a\;b=1$, $b^2 = 2$ and $a\;c = 4$ to obtain $p = x + 2y + 4z$. How can I tell Mathematica to do that? I dont know how to ...
5
votes
1answer
633 views

Factoring a two variable polynomial in a special way

Let $$f=x^9-x^6+4x^5y+2x^3y^2-y^4$$ I would like to factorize $f$ into form: $$(y-F_1(x))\cdots(y-F_k(x))$$ over complex numbers. How can I do it with Mathematica?
4
votes
3answers
974 views

Finding parameters making real part of eigenvalues vanish

I have the following $\;3\times3$ matrix: $\left( \begin{array}{ccc} 0.04 -0.4 b & 0 & 0.04 -0.4 b \\ 0 & -0.08-1.2 b & -0.06-0.9 b \\ 1.04 -0.4 b & 2.08 -0.8 b & 0 \end{...
24
votes
11answers
7k views

Defining a function that completes the square given a quadratic polynomial expression

How can I write a function that would complete the square in a quadratic polynomial expression such that, for example, CompleteTheSquare[5 x^2 + 27 x - 5, x] ...
17
votes
4answers
5k views

How do I find the degree of a multivariable polynomial automatically?

I have a very simple question which appears not to have already been answered on this forum. Is there built-in functionality that returns the degree of a multivariable polynomial? For example if the ...
15
votes
5answers
3k views

How to get exact roots of this polynomial?

The equation $$ 64x^7 -112x^5 -8x^4 +56x^3 +8x^2 -7x - 1 = 0 $$ has seven solutions $x = 1$, $x = -\dfrac{1}{2}$ and $x = \cos \dfrac{2n\pi}{11}$, where $n$ runs from $1$ to $5$. With ...
14
votes
2answers
537 views

Does NRoots own an abstract counterpart? If not, can we write one?

We know when solving linear algebra equations, despite its abstract syntax, LinearSolve is much faster compared to Solve: ...
8
votes
2answers
2k views

Finding the coefficient of a certain power in a generating function

I want to compute $(t^1+\dots +t^5)(t^2+\dots+ t^6)(t^3+\dots +t^9)$ and find the coefficient of $t^{15}$ for example. $t$ is an indeterminate Now in Maple this is simply as ...
7
votes
1answer
1k views

How to do the polynomial stuff over finite fields extensions fast?

This question is raised from the problem of package FiniteFields being very slow (please, see the corresponding question): I have had an evidence that Mathematica ...
4
votes
2answers
689 views

Polynomial expansion of operator

I am new to Mathematica, I am trying to generate the polynomial function of a operator. So for example, the operator $L $ is $\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} $, and I want ...
18
votes
2answers
1k views

Expressing a polynomial as a sum of squares

I encountered a degree 4 polynomial in 8 variables f(a1,a2,a3,a4,b1,b2,b3,b4) that I suspect can be written as a sum of squares. While sostools in MATLAB would find ...
17
votes
6answers
4k views

Create a polynomial of a given degree

In Mathematica, how can I create a polynomial function in given variables of a given degree with unknown coefficents? That is, I am looking for a function ...
6
votes
3answers
1k views

How to get a list of monomials of a polynomial without coefficients?

Giving a polynomial, say a x^2 + b x y + c y^2 MonomialList[a x^2 + b x y + c y^2, {x, y}] just gives ...
12
votes
4answers
918 views

How to replace every possible $A+B$ and $AB$ in expansion of $(A+B)^{10}-A^{10}-B^{10}$ with $x$ and $y$, respectively?

I would like to replace every possible $A+B$ and $AB$ in expansion of $(A+B)^{10}-A^{10}-B^{10}$ with $x$ and $y$, respectively. How to do it with the simplest code in Mathematica? For example, \...
9
votes
4answers
2k views

How to collect terms with positive powers in polynomial

I am trying to collect all terms with non-negative powers of $x$ in polynomials like $\frac{1}{x^2}\left(a x^2+x^{\pi }+x+z\right)^2$ First expand the polynomial ...
8
votes
3answers
1k views

Build a Companion Matrix of a Polynomial?

The eigenvalues of a square matrix $A$ are the roots of its characteristic polynomial $\chi(\lambda)$. Conversely, if we have a monic polynomial $p(\lambda)=a_0 + a_1 \lambda + \cdots + a_{n-1}\lambda^...
6
votes
1answer
107 views

Loss of accuracy in orthogonalisation of polynomials using Orthogonalize

Context As a mean to understand the growth of structure in the universe, I am interested in characterising the curvature of random fields such as this one: For this purpose I start with a PDF of the ...
4
votes
4answers
3k views

Calculating Taylor polynomial of an implicit function given by an equation

I'd like to write a procedure that will take an equation: F(x,y,z) = 0 chosen variable: x a point: ...
2
votes
2answers
798 views

How to know form of plotted Bézier function

Simple scenario is to see the Bézier function, but how to know which polynomial approximate it? ...
10
votes
3answers
564 views

Wrong results from NSolve on coupled polynomials. WorkingPrecision -> Automatic fails

OP UPDATE: I received an email from WR on 1-18-2016: "...It does appear that the NSolve function is not behaving properly in this case and I have forwarded an incident report to our developers with ...
4
votes
2answers
292 views

HornerForm of polynomials in terms of E^(i x)

I want to know how to get the HornerForm of the following expression in terms of E^(I x): ...
3
votes
1answer
302 views

Collecting factors of polynomials

First up, I'm new to Mathematica and MathematicaSE, so please correct me if I'm doing anything wrong. My problem is the following: I have a bunch of complicated polynomials in $3$ variables $x_1,x_2,...
3
votes
5answers
7k views

The plot of roots of polynomials

I have polynomial equation like Tribonacci Polynomials for example: $T_3(x)=x^4+x$. After finding the roots of this polynomial, I want to show these roots in the complex plane. I have tried lots of ...
15
votes
3answers
776 views

What are Root objects with multiple polynomials?

In Mathematica 9 a new flavor of Root object with multiple polynomials was introduced. For example, ...
16
votes
5answers
3k views

Computing the genus of an algebraic curve

How-to compute the genus of an algebraic curve in Mathematica ? In my case the algebraic curve is explicitly defined by a polynomial.
17
votes
4answers
6k views

Is there a way to Collect[] for more than one symbol?

Oftentimes you find yourself looking for polynomials in multiple variables. Consider the following expression: a(x - y)^3 + b(x - y) + c(x - y) + d as you can ...
7
votes
6answers
7k views

Series expansion in terms of Hermite polynomials

I am trying to expand a polynomial in terms of orthogonal polynomials (in my case, Hermite). Maple has a nice built-in function for this, ChangeBasis. Is there a ...
11
votes
2answers
7k views

How to define a polynomial/function from an array of coefficients?

I have the coefficients of my desired polynomial in an array CoefArr (I'm new to mathematica, so I think of everything as arrays, it is actually a list I believe) starting with the constant at index 1....
13
votes
3answers
1k views

Factor a polynomial over the reals

I can't seem to find a function in Mathematica that factors a polynomial over the reals. Obviously, Factor doesn't work since it works over the integers, over $\mathbb{Z}_p$, or some algebraic fields ...