Questions tagged [polynomials]
Questions on the functionality operating on polynomials
983
questions
53
votes
6
answers
12k
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 ...
- 1,422
53
votes
4
answers
21k
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 ...
- 4,503
50
votes
4
answers
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:
...
- 1,493
35
votes
5
answers
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 ...
- 19.7k
25
votes
11
answers
8k
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]
...
- 413
24
votes
2
answers
2k
views
Code I get from wolfram isn't working in mathematica
I need to perform the factor as shown here
factor(s^5+32s^4+363s^3+2092s^2+5052s+4320)
https://www.wolframalpha.com/input/?i=factor(s%5E5%2B32s%5E4%2B363s%5E3%...
- 363
22
votes
7
answers
19k
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
...
- 2,623
22
votes
4
answers
4k
views
How to keep Collect[] result in order?
For example,
Collect[(1 + x + Cos[s] x^2)^3, x]
gives the result
...
- 17.1k
22
votes
4
answers
17k
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 ...
- 413
22
votes
1
answer
420
views
Undocumented fourth parameter of Collect; how long has it been there?
While tinkering with How to get a list of monomials of a polynomial without coefficients? on a whim I tried a fourth argument in Collect and found:
...
- 272k
21
votes
4
answers
3k
views
Neural Network for polynomial fit
I'm trying to build up a neural network with Mathematica 11.0, that should fit data which behaves like a polynom of third order.
I thought that an NN with one or two hidden layers can fit any function,...
- 211
21
votes
2
answers
2k
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 ...
- 311
20
votes
0
answers
336
views
Changes to Coefficient function in v10.2
In the version Mathemaica 10.4, I am very surprised that the core function Coefficient has changed, e.g.,
...
- 1,247
19
votes
2
answers
745
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} {...
- 8,958
19
votes
2
answers
6k
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 ...
- 535
18
votes
5
answers
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 ...
- 34k
18
votes
5
answers
6k
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 ...
- 527
18
votes
4
answers
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 ...
- 2,623
17
votes
4
answers
785
views
What is Mathematica's equivalent to Maple's collect with distributed option?
Given a polynomial in $x,y$, I want to collect on $x,y$ and any products of these also. As given using Maple's collect with the distributed option.
Currently Mathematica will collect on $x$ then ...
- 144k
17
votes
6
answers
6k
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 ...
- 313
17
votes
2
answers
2k
views
Solving quintic in radicals
I need to find an explicit expression in radicals for the real root of the quintic equation
...
- 7,213
17
votes
1
answer
892
views
Mathematica incorrectly giving zero for partial derivative
Bug introduced in 9.0 and fixed in 11.1
Mathematica is incorrectly reporting that the partial derivative of a certain expression is zero.
I try to compute the following:
...
- 271
16
votes
5
answers
4k
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.
- 263
16
votes
5
answers
4k
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 ...
- 4,503
16
votes
2
answers
663
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:
...
- 66.2k
15
votes
2
answers
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 ...
user6193
15
votes
2
answers
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...
<...
- 1,017
15
votes
3
answers
849
views
What are Root objects with multiple polynomials?
In Mathematica 9 a new flavor of Root object with multiple polynomials was introduced. For example,
...
- 395
14
votes
2
answers
783
views
Finding the number of odd quintinomial coefficients
I am searching for the number of odd coefficients of
$\qquad (x^4 + x^3 + x^2 + x + 1)^n$
for arbitrary $n$.
It took some hours to compute the result for $n=12207$. There are $16333$ odd ...
- 6,697
14
votes
4
answers
990
views
Why Mathematica can not factorize polynomials over algebraic fields?
I noticed that a factorization over algebraic fields is useless in Mathematica. Here is the example over the field containing I*Sqrt[3]:
...
- 181
14
votes
2
answers
777
views
How can I compute the representation matrices of a point group under given basis functions?
Take the $C_{3v}$ point group for example:
...
- 605
14
votes
1
answer
195
views
What's going on here? Some kind of rationalization "under the covers"?
Observe:
eq = (.25 a + .5 b + .25 c);
CoefficientRules[eq^2]
CoefficientRules[eq^2 // Expand]
results in
{{2, 0, 0} -> 1/16, {1, 1, 0} -> 1/4, {1, 0, 1} -...
- 25.8k
13
votes
3
answers
2k
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 ...
- 349
13
votes
3
answers
11k
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 ...
- 233
13
votes
1
answer
450
views
ToNumberField won't recognize Root as an explicit algebraic number
Bug fixed in 10.0.0
In Mathematica 9.0.1, it appears that ToNumberField will not always recognize a Root object as an explicit ...
- 1,382
13
votes
1
answer
1k
views
Rearranging a Polynomial
In Mathematica 8.04 on Windows, I want to display a formula in standard textbook format. The formula is the variance of an $N$-security portfolio. For two securities it is:
$$w_1^2\sigma_1^2+w_2^2\...
- 1,449
12
votes
4
answers
1k
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,
\...
- 757
12
votes
5
answers
8k
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....
- 123
12
votes
4
answers
1k
views
Any efficient way to make complete homogeneous symmetric functions in Mathematica?
We do have elementary symmetric functions, SymmetricPolynomial[k, {x_1, ..., x_n}] .
But I didn't find complete homogeneous symmetric functions.
The induction ...
- 1,019
12
votes
2
answers
791
views
Surprises simplifying simple polynomials
I came across some somewhat surprising behavior of Simplify today, on something very simple. Let's take two cubic polynomials that we know have the same value:
<...
- 18.5k
12
votes
2
answers
697
views
How to deduce a generator formula for a polynomial sequence?
Consider a polynomial sequence $\{p_n\}$ generated by some (simple) rule:
$$
\begin{array}{l}
p_1(x)=x \\
p_2(x)=2 x-x^2 \\
p_3(x)= x^3-3 x^2+3 x \\
p_4(x)=-x^4+4 x^3-6 x^2+4 x \\
p_5(x)= x^5-5 x^4+...
- 12.7k
12
votes
1
answer
393
views
Using Mathematica to find an alternative continued fraction for $\zeta(5)$
Given the Riemann zeta function $\zeta(n)$.
I. $x=\zeta(3)$
Using Euler's continued fraction formula, we can form $\zeta(3)$'s cfrac as,
$$Ax+B = \cfrac{1}{v_1 - \cfrac{1^6}{v_2 - \cfrac{2^6}{...
- 447
11
votes
3
answers
4k
views
Get the coefficient matrix from a quadratic form
Suppose I have a quadratic form of
qf = a x^2 + b y^2 + c z^2 + 2 d x y + 2 e x z + 2 f y z
How can I easily to get the symmetric matrix A, such that $X^TAX=qf$?...
- 1,235
11
votes
6
answers
2k
views
Can Mathematica tell me if a polynomial has all real roots?
I am trying on this polynomial,
...
- 1,181
11
votes
3
answers
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 ...
- 1,141
11
votes
3
answers
9k
views
First positive root
Simple question but problem with NSolve.
I need help how to extract first positive root?
For example:
...
- 1,099
11
votes
2
answers
677
views
11
votes
3
answers
3k
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] ...
- 113
11
votes
4
answers
1k
views
Mathematica function equivalent to MATLAB's residue function (partial fraction expansion)
I am looking for a Mathematica function equivalent to MATLAB's residue function.
If there is no Mathematica equivalent, I would like to write a function that given ...
- 155
11
votes
1
answer
1k
views
How can I plot a Chebyshev spiral?
The Chebyshev polynomials T_n of the first kind are a certain set of orthogonal polynomials. They can be defined by T_n(cos(x))=cos(nx), the first of them are
T_0(x) = 1
T_1(x) = x
T_2(x) = ...
- 113