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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Representing as sum of squares of polynomials

Show that the polynomial $x^4y^2+y^4z^2+z^4x^2-3x^2y^2z^2$ cannot be written as the sum of squares of polynomials over $\mathbb{R}$ in $x, y, z$. I could not make any progress/significant ...
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Linear forms in $\Bbb Z[x]$

Supposing you have a sextic $A(x)\in\Bbb Z[x]$ $$A(x)=G_1(x)H_1(x)+G_2(x)H_2(x)+G_3(x)H_3(x)$$ where each $G_i(x)\in\Bbb Z[x]$ at $i\in\{1,2,3\}$ is a quartic satisfying ...
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Why do I get homogenizations of polynomials by trying to find roots in $\mathbb Q$.

I noticed that if I have a polynomial equation in, say $x$ that needs to be solved in $\mathbb Q$, one tactic is to substitute $x=y/z$ where $y$ and $z$ are coprime integers, but then after clearing ...
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Smoothness in cyclotomic versus complex fields?

Say we have a polynomial in a cyclotomic field; in particular, an n-th cyclotomic field, where n is the order of the polynomial's symmetry group. If we know the polynomial is smooth over this field, ...
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How to scale polynomial coefficients for root-finding algorithms?

I've implemented the Jenkins Traub algorithm in c++ (Github repo). While the majority of the solutions work well, it seems that a small portion of the roots are unstable. Here is a link to a ...
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Help with rearranging a large polynomial

I have an expression $$\frac{(-ab)x^2+(a^2+b^2)xy+(-ab)y^2}{2(a-b)^2(0.5(a+b))^2}$$ (a and b are constants) and I need to rearrange it so that it is in the form f(x)*f(y). I am totally lost, and ...
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4answers
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Determining if function odd or even

This exercise on the Khan Academy requires you to determine whether the following function is odd or even f(x) = $-5x^5 - 2x - 2x^3$ To answer the question, the instructor goes through the following ...
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Splitting a sextic into cubics

I have a sextic in $\Bbb Z[x]$ with very large coefficients that I know splits into two cubics in $\Bbb Z[x]$. What is the best way to do find the cubics?
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Solving nonlinear system

I have the following nonlinear system $$\begin{cases} y_1 = \frac{x_1}{\sqrt{x_1^2+x_2^2+x_3^2}} \\ y_2 = \frac{x_2}{\sqrt{x_1^2+x_2^2+x_3^2}} \\ y_3 = \frac{x_3}{\sqrt{x_1^2+x_2^2+x_3^2}} \end{cases} ...
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1answer
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Integer polynomials and the integration on the unit circle

Let $f (x) $ be a polynomial in $\mathbb {C}[x] $. Then, consider the value $I_f $ defined as $I_f:=\displaystyle\int_0^{\frac {\pi}{2}} |f (e^{ix})|^2 dx$. We can define this on any complex ...
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5answers
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Is it possible to factor a quadratic equation when $a$, $b$, and $c$ are all equal?

I have the equation $4x^2+4x+4$ to factor. I know that need to start with $$(2x \quad )(2x \quad )$$ to make $4^2$, but I can't seem to factor the rest of the way. What should I do?
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General Fiber in Positive Characteristic

It is well-known that a complex polynomial, considered as a function $f:\mathbb{C}^n\to\mathbb{C}$, is a fiber bundle over a cofinite set of "atypical values" which include the singular values of $f$. ...
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Reduce multivariate polynomials by known roots?

Consider three multivariate polynomials $p_1(x,y,z)$, $p_2(x,y,z)$ and $p_3(x,y,z)$ with $x,y,z\in\mathbb{C}$. Imagine that the set of polynomials above is constructed such that they have exactly $6$ ...
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Is there a polynomial $p$ such that it is bijective and $ p: \mathbb{Q}^n \rightarrow \mathbb{Q}$ for $ n>1$ ??

Let us define a polynomial $p$ defined as follow $$p: \mathbb{Q}^n \rightarrow \mathbb{Q}.$$ I acrossed this question in mind. Is there a polynomial $p$ such that it is bijective and $p: ...
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1answer
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Degree of Polynomial in Centered Moments of Gamma$(n,1)$

I'm interested in the degree of the polynomial in $n$ of the expression for the $k$-th central moment $$ E((X_n - n)^k) $$ where $X_n$ is a Gamma$(n,1)$ random variable, that is, the sum of $n$ ...
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A question in matrix polynomial.

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
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3answers
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$A$ is a $n \times n$ matrix over $\mathbb{R}$ such that $A^2+A+5I=0$. Find the characteristic polynomial of the matrix $A$. [on hold]

$A$ is a $n \times n$ matrix over $\mathbb{R}$ such that $A^2+A+5I=0$. Find the characteristic polynomial of the matrix $A$. it is a question from a test i had yesterday and this is how it was ...
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1answer
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Is $P(X,Y)=a + aY + (b+cX^2)Y^n \in \mathbb Z [X][Y]$ irreducible?

I am considering the polynomial $a + aY + (b+cX^2)Y^n\in \mathbb Z [X][Y]$, with $n$ even and $a,b,c$ non zero integers. Is this polynomial irreducible or not?
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Factoring bivariate quadratics with real coefficients (for high school students).

I was tutoring a Year 10 student last night (he's learning about quadratics). Unfortunately, we ran into a class of problems that I couldn't explain how to solve (beyond simply guessing and checking), ...
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limit of homogeneous polynomials as the order approaches infinity?

Let $f_n(x,y)=\sum a_ix^iy^{n-i}$ be a sequence of homogeneous polynomials with fixed coefficients $a_i$, is it possible to make $f_n$ converge as $n\rightarrow\infty$ after possible renormalization? ...
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How do I fit $f(x) = \exp(a+bx+cx^2 +dx^3)$ to two points? $x, f(x)$ and $f'(x)$ are known.

In the past I've fit polynomials by solving the set of equations. I can fit $f(x) = \exp(a+bx)$ to point $A$ and $B$ where I know $x$ and $f(x)$ for both points. If I want to fit to a specific ...
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Polynomial Interpolation When $y_i$'s are Permuted

Recall, if we have a $d$-degree polynomial $f$, evaluate it at $\textbf{x}=(x_1,\ldots,x_n)$ we would get $\textbf{y}=(y_1,\ldots,y_n)$, where $f(x_i)=y_i$ and $d+1 \leq n$. The reverse is also true, ...
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1answer
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Is every algebraic integer a sum of roots of $x^n - a$?

A complex number is said to be an algebraic integer if it is a root of a monic polynomial with integer coefficents. For example any root of the polynomial $x^n - a$ for $a \in \mathbb{Z}$ is an ...
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Proof for two monic polynomial gcds, $d$ and $d_0$, if $d|d_0$ and $d_0|d$, then $d=d_0$

This is an extension to this, that is covered in my higher linear algebra course. I know if $d$ and $d_0$, both $\in \mathbb{F}[x]$ are monic and gcds of some polynomials $g$ and $f$ in ...
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solve another system of three equations [on hold]

I have: $x=\dfrac{-.5b-.5c+.25d}{b+c+d}$ $y=\dfrac{.5b\sqrt{3}+.5c\sqrt{3}+.25d\sqrt{3}}{b+c+d}$ $z=b+c+2d$ I need help moving the $b$, $c$, and $d$ to the Left-hand-side; and moving the x, y, and ...
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1answer
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Minimize distance between polynomials, of a certain form, with Laguerre polynomials

A typical problem that I may encounter on an upcoming test looks like this: Find the polynomial $P(x)$ of a degree less than or equal to three that minimizes $$\int_0^\infty (x^4 - ...
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How to find $p\in \Bbb C[X]$ given $p(p(X))$

Assume you're given $p(p(X))$ in the form $$p(p(X))= \sum_{iā‰„0} a_i X^i$$ Is there any quick algorithm to retrieve $p$? What can be said about the degree of $p(p)$ I think it's twice the degree of ...
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Polynomials with range containing an arithmetic progression

Can I find a polynomial in a second degree in two variables from the values of which can be found an infinite arithmetic progression? Thank you!
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Eigenvectors of the companion matrix

Suppose one has an Hermitian square matrix $A$ with $p$ is the characteristic polynomial $$ p(x)= a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n ~, $$ and define the companion matrix of $p$ as $$ ...
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1answer
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what is f(x) < 0 asking for?

I'm trying to answer a question that says, State where $f(x)<0$ using any correct notation and I do not know what it is asking for. The question provides me a graph going from quadrant 2 to 4, and ...
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Zero locus of 2-variate real polynomial are smooth curves

This seems like it should be an easy question, and probably already has already had answer in advanced mathematics, but I only know some basic calculus, so I would like to know how do I go about doing ...
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2answers
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$\frac{y-b}{r}=\frac{y}{s}$ to $y$ for finding the closest point on a line, from a point.

$$r=sy^2-sby$$ How do I get $y$ on one side? Originally I had: $\dfrac{y-b}{r}=\dfrac{y}{s}$
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Proof that the Runge Phenomenon occurs

Is there such a proof that states that the Runge Phenomena will always occur when interpolating with higher order polynomials or is this just observed empirically?
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43 views

For which elements $t$ in a finite field $\mathbb{F}_{p^n}$ is $t^2 - 4$ a square?

That is, how to characterize the elements $t \in \mathbb{F}_{p^n}$ for which there exists $x \in \mathbb{F}_{p^n}$ such that $t^2 - 4 = x^2$?
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1answer
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Show that $(t^m-1)/(t^n-1)$ is a square if and only if $(\exists s \in \mathbb{Z})\ m=np^s$

I want to show the following lemma: Assume that the characteristic of the field $F$ is $p$ and $p>2$. Then $(t^m-1)/(t^n-1)$ is a square in $F[t, t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in ...
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Polynomial Interpolation and Data Integrity

This question is about polynomial interpolation and security. Please consider a scenario where we have a polynomial $f$, one of whose roots is $a$. We evaluate it at some ...
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1answer
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Determining polynomial values

The polynomial has been edited to include the "x" term $R(x)= x^4+Ax^3+Bx^2+10x-1$ has a remainder of $-15$ when divided by $x+1$ and a remainder of $39$ when divided by $x-2$. Determine $A$ and ...
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Equality regarding Bernstein polynomials

The Bernstein polynomials are defined like this: $b_k(m,x)= {{m}\choose{k}} x^k(1-x)^{m-k}$, if $k<m$ I want to prove that $\sum\limits_{j=k}^m b_j(m,x) = m {{m-1}\choose{k-1} }\int\limits_{0}^x ...
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How would one define polynomials over the projective line $P_K^1$

May $K$ be a field. If I set $\varXi=(X:Y)$ as a "projective variable" and "projective coefficients" $a_k=(x_k:y_k)\in P_K^1$ - may I then write a polynomial map $P_K^1\longrightarrow P_K^1$ in a form ...
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1answer
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Functional equation: Finding $f(100)$

A polynomial of degree 98 such $f (k)=1/k$ for $k=1,2,3...,98,99$ exists. How to find $f(100)$? What are the possible methods ?
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Decomposition of a homogeneous polynomial

Let $k$ be a field. Suppose I have a homogeneous polynomial $f$ in $k[x,y,z]$. If $f$ is reducible, does it always decompose as a product of homogeneous polynomials? Thanks!
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Does every non trivial variety in $\mathbb{R}^n$ have empty interior?

By this question, we know that a non-trivial affine variety in $\mathbb{C}^n$ has empty interior. But the argument uses the (strong) fact that a holomorphic function vanishing in a non empty set $U$ ...
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1answer
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Factoring $x^{15}āˆ’1$ into irreducible polynomials over $\mathrm{GF}(2)$

Factorize $x^{15}āˆ’1$ into irreducible polynomials over $\mathrm{GF}(2)$ The answer is $$(x+1)(x^2+x+1)(x^4+x+1)(x^4+x^3+1)(x^4+x^3+x^2+x+1)$$ but how would I find this? Please help.
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3answers
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How does determining the area of rectangle relate to binomial multiplication?

So using the strategy to determine the area of the large rectangle I simply did $10\times10, 10\times2, 10\times4, 2\times4$ to get $168\mathrm{cm}$ total. The next question goes on to ask how ...
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1answer
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Number of polynomial factors of $a^n-b^n$?

This is a number theoretical problem that I discovered myself. Let $f(n)$ be the number of factors of $a^n-b^n$ with integer coefficients when its completely factored. For example: $f(1)=1$, because ...
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2answers
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finding factors of a polynomial

In the math problem in the attached image, it explains how to find the factors of a polynomial whereby every possible factor of the function is of the form p/q, where "p is a factor of the constant ...
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Splitting even degree polynomials

I have an octic equation (degree $8$) and a sextic equation (degree $6$) in $\Bbb Z[x]$ with very large coefficients (size several hundred bits) that I know splits into two quartics and two cubics ...
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2answers
67 views

Find all the possible real values for $a,b,c,d$.

Let pairs $(a,c)$ and $(b,d)$ be roots of the equations $x^2 + ax - b = 0$ and $x^2 + cx + d = 0$ respectively. Find all possible real values for $a,b,c,d$.
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2answers
45 views

How to split a quartic into two quadratics?

I have a quartic in $\Bbb Z[x]$ with very large coefficients that I know splits into two quadratics in $\Bbb Z[x]$. What is the best way to do find the quadratics?
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1answer
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Inverting power sum of symmetric polynomial

Suppose I have a set of power sum symmetric polynomial as $$S_p =\sum_i^N x^p_i ~~;~~~~~~~~p=\{1,N\}$$ and I have N of them $\{S_1...S_N\}$ Question is given this, can we find ${x_n=F(\{S_p\})}$? ...