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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Solve $x+\frac{2}{y}=3,y+\frac{2}{z}=3,z+\frac{2}{x}=3 $ in reals

find answers of this system of equations in real numbers$$ \left\{ \begin{array}{c} x+\frac{2}{y}=3 \\ y+\frac{2}{z}=3 \\ z+\frac{2}{x}=3 \end{array} \right. $$ Things i have done: first i ...
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Question about Horners rule algorithm

I am studying Horner's rule I have a question about an algorithm I found here . I understand that the rule allows you to break down polynomials in to monomials to solve them more easily, so that for ...
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3answers
66 views

Prove $\frac{a}{(b-c)^2}+\frac{b}{(c-a)^2}+\frac{c}{(a-b)^2}=0$ if $\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0$

if $a,b,c$ are real numbers and $$\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0$$ Prove $$\frac{a}{(b-c)^2}+\frac{b}{(c-a)^2}+\frac{c}{(a-b)^2}=0$$ things i have done: using the assumption i ...
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1answer
86 views

A polynomial agreeing with a function and its derivatives

If we want $$p(x_i)=a_i, \qquad x_1 < \dotsb < x_{n+1},$$ then there is a unique polynomial of degree $\leq n$ that accomplishes this (Lagrange interpolation). If we want $$p(x_i)=a_i, \qquad ...
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Finding roots of polynomials of arbitrary degree

I asked this question on MO, but it has been tagged off-topic. Is there any analogue of the method for expressing roots of polynomials of degree $5$ with elliptic and $\eta$-functions that ...
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2answers
69 views

How many roots has the equation?

Let $f(x)=x^3-3x+1$. How many roots has the equation: $f(f(x))=0$? I tried to solve it graphically and found 7 roots. If there exists analytical solution?
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Non-real coefficients, but real roots

Say I want to provide an example of a polynomial with non-real coefficients, but with real roots. A trivial example to provide is $P(x)=ix-i$, which has non-real coefficients but a real root $x=1$. ...
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32 views

How to find factor when the polynomial is not given

For the question " If $f(x)$ is a polynomial with constant term $10$ having a factor $(x-k)$ where $k$ is an integer, then find the possible value of $k$", the options given are $-20, 20, 8$ and $5$. ...
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Show that a function is a solution to differential equation [duplicate]

I have a homogenous differential equation $a_0 y'' + a_1 y' + a_2 y = 0$ I know that $\lambda_0$ is a double root in characteristic polynomial. Now I have to show that $y(t) = t e^{\lambda_0 t}$ is ...
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1answer
34 views

What is the formal definition of polynomial ring of several variables?

Let's consider a polynomial ring of single variable. One can define them informally by saying $P(X)=\sum_{i=1}^n a_n X^n$ while $X$ is an indeterminate variable. However, since mathematics is based ...
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Solve $ x^2+y^2=4, z^2+t^2=9, xt+yz=6 $ in integers

find answers of this system of equations in integers$$ \left\{ \begin{array}{c} x^2+y^2=4 \\ z^2+t^2=9 \\ xt+yz=6 \end{array} \right. $$ things I have done: we can observe that ...
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2answers
31 views

Prove $a^4+b^4+(a-b)^4=c^4+d^4+(c-d)^4$ if $a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$

if $a,b,c,d$ are positive real numbers and $$a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$$ Prove $a^4+b^4+(a-b)^4=c^4+d^4+(c-d)^4$ Things i have done: from assumption $a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$ I ...
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1answer
27 views

continuity of polynomial of two variables

We know that polynomial functions are always continuous. The proof which I did was only for single variable polynomial. What about the polynomials in two variables? Can we say a polynomial of two or ...
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171 views

How do we check if a polynomial is a perfect square?

Often we come across polynomial diophantine equations of the form $y^2 = x^2 + x+ 1$ and there are two ways to disprove the existence of solutions to such equations: 1) By bounding between ...
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Factoring $a^4(b-c)+b^4(c-a)+c^4(a-b)$

I was solving the question that wanted to factor $a^4(b-c)+b^4(c-a)+c^4(a-b)$. My idea was to factor a $(c-a)$ in first step.So $$b(a^4-c^4)+ac(c^3-a^3)+b^4(c-a)=a^4(b-c)+b^4(c-a)+c^4(a-b)$$ ...
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39 views

Condition for zeros of a polynomial in Unit Disk

Consider a polynomial in $\mathbb{C}$ with complex coefficients, $\lambda^2+p\lambda+q$ where both $p$ and $q$ are complex numbers. I am looking a for a condition of $p$ and $q$ such that the zeros ...
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62 views

prove that a polynom is zero

Let $m \in\mathbb N$. Define a polynom $P$ by: $$ P(x)=\sum_{k=0}^{m+1} \binom{m+1}{k}(-1)^k (x-k)^m $$ Prove that $P(x)\equiv 0$. I tryed to use taylor polynomials, finding roots, but it did not ...
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1answer
26 views

Sum of Lagrange basis polynomials

Let $L_i(x)$ be Lagrange basis polynomials for $n+1$ points $(x_0,y_0),\ldots, (x_n,y_n)$. How do you prove that $\sum_{i=0}^n (x-x_i)^pL_i(x)=0$ for $p\leq n$?
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Multivariate Appoximation

I have a mathematical model for a complex system which I would like to approximate it. My idea is to run this complex model once and produce some outputs, and then fit a polynomial for these outputs. ...
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Prove $a^2+b^2+c^2=\frac{6}{5}$ if $a+b+c=0$ and $a^3+b^3+c^3=a^5+b^5+c^5$

if $a,b,c$ are real numbers that $a\neq0,b\neq0,c\neq0$ and $a+b+c=0$ and $$a^3+b^3+c^3=a^5+b^5+c^5$$ Prove that $a^2+b^2+c^2=\frac{6}{5}$. Things I have done: $a+b+c=0$ So ...
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What's the difference between $(ap)(x)$ and $ap(x)$ if $p$ is a polynomial and $a$ is constant?

Would it be correct to say $(ap)(x)$ is a polynomial $ap$ evaluated at $x$ and $ap(x)$ is a polynomial $p$ evaluated at $x$ with all its coefficients multiplied by $a$?
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2answers
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For which $p$ and $q$ polynomials $x^q-1$ and $(x+1)^q-1$ are coprime in $F_p[x]$?

It easy to prove that polynomials $x^q-1$ and $(x+1)^q-1$ are coprime in $\mathbb{Q}[x]$ if $(q,6)=1$, since they don't have a common zero in $\mathbb{C}$, this can be seen geometrically. My question ...
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5answers
298 views

Evaluating a polynomial of degree 4, given some values of the polynomial

If $p(x)$ is a polynomial of degree 4 such that $p(2)=p(-2)=p(-3)=-1$ and $p(1)=p(-1)=1$, then find $p(0)$.
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Find the condition on b so that 6th degree polynomial has at least three real roots

Most of the questions on this site ask: Given a polynomial, how to find the number of real roots. My question is: given a 6th degree polynomial $P_b(x)$: (b lies between 1 and 4) ...
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How to prove that all zeros of the complex polynomial $P(z)$ lie in the closed unit disk $|z| \leqslant 1$?

I want to know how to prove that all zeros of the polynomial $P(z)$ lie in the closed unit disk $|z| \leqslant 1$. Where $$P(z)=z^{n+1}+\frac{2(n+1)\cos\alpha}{n+2}z^{n}+\frac{n}{n+2}z^{n-1}+\frac{2 ...
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1answer
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Solving for a single variable in a quadratic system

I have a quadratic system of $n$ equations that looks like: $$ A_{ij}x_{j}y + B_{ij}x_{j}=0 $$ For $i=0...n$, where $A_{i,j}$ and $B_{ij}$ are integer matrices and sums over $j$ are implied. Is there ...
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How to expand quadratic equations in Octave/Matlab?

I have some column vectors, and I put them in a row, so I have $[a \, b \, c]$. Now I want to get a matrix of the form $[a^2 \, b^2 \, c^2 \, ab \, ac \, bc]$. It is like expanding $(a+b+c)^2$, but ...
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Self-contained formal polynomial reference

In the forward to the third edition of his Undergraduate Algebra, Lang mentions: A new section in Chapter IV gives a complete account of the Mason-Stothers theorem about polynomials, with Noah ...
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Polynomial transformation of the roots of another irreducible polynomial.

Suppose I have some monic irreducible polynomial $g(x)$ in $\mathbb{Z}[x]$ with distinct roots $r_1,r_2,\dots,r_n$. Suppose $f(x)$ is some other polynomial, not necessarily irreducible. Is there ...
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Find prize per unit that will maximize profit at a given $x$-value

Struggling while reviewing my old math books. The problem has a prize-function and wants to know how the prize-per-unit should be chosen to maximize the profit at $\mathbf{x=160}$. First I look ...
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Does $f(n,z)$ have $2^n$ distinct fixpoints $z$ for all $n$?

Let $f(z)$ be a given degree $2$ polynomial. Let $n$ be a positive integer. Let $f(1,z) = f(z)$ and $f(n,z)= f(1,f(n-1,z))$. How to decide if $f(n,z)$ has $2^n$ distinct fixpoints $z$ for all $n$ ?
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Achieving a polynomial that maps from $GF(p^q)$ to {0,1} with the same probability

I am using an arithmetic circuit, which can compute polynomials over the field $GF(p^q)$. I need a polynomial that maps any element from the field to an element from $\{0,1\}$, I need that the range ...
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150 views

Factoring the following polynomials

Factorize the following polynomial: $t^3 -9t +8$ $t^6 -91t^2 +90$
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Constructing a polynomial bump function

Proposition: Suppose $f$ is continuous and $\int_a^bf(x)x^ndx = 0$ for all $n$. Then $f$ is zero on $[a,b]$. This can be proven by uniformly approximating $f$ with polynomials via the Weierstrass ...
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A complex polynomial in $z$ and $\bar z$ contains no terms with $\bar z$ if and only if its $\bar z$-derivative is zero

I am struggling with this exercise: Let $p(x,\bar{z})=\sum a_{lm}z^l\cdot\bar{z}^m$ be a polynomial in $z$ and $\bar{z}$ (so only finitely many $a_{lm}$ are non-zero). Show that p contains no ...
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Solving a Bernstein Polynomial for 3D space (trivariate)

I'm writing a piece of software and need to deform points in 3D space by a set of control points. After some searching I found this paper on how to do it. The summary is 'The deformation function is ...
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Natural cubic spline interpolation - check and suggest better way

I was given the following interpolation nodes: $(0,10),(\frac{1}{2},8),(1,5),(2,2),(3,1)$ and I was asked to find the natural cubic spline interpolation between every 2 points. I want to show you ...
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Square-free factorization of polynomials over finite fields

For any $f\in\mathbb{F}_q[X]$, I want to derive an algorithm which computes a factorization $$f=\prod_{i=1}^kf_i^i\tag{1}$$ with square-free polynomials $f_i$. My Ideas: If $f'=0$, we're done ...
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How to prove that all primitive polynomials are irreducible

Let $F$ be a finite field, and $F[X]$ set of all polynomials in $F$, how to prove that: why all primitive polynomials $\;$ $f \in F[X]$ $\;$ must be an irreducible. Note: Polynomial primitive is an ...
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Why do we interpolate - no guarantee of success

this is somewhat of a general question about interpolation, I don't fully understand how can we be confident that our approximation is good, even if we know a lot of points. An example would be: ...
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8answers
202 views

How do I solve $x^5 +x^3+x = y$ for $x$?

I understand how to solve quadratics, but I do not know how to approach this question. Could anyone show me a step by step solution expression $x$ in terms of $y$? The explicit question out of the ...
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Construction of a polynomial of degree 4 with some conditions

Exercise Let $P(x)$ be a polynomial of degree $4$, the question is : Find this $P$ such that : The coefficient oh highest degree is $1$ P is divisible by $x^2+x+1$ The rest of the division of $P$ ...
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Proving degree $n$ have at last $n$ roots in $F_q[X]$

How to prove that in $F_{q}[X]$ of degree $n$, have $n$ roots?
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66 views

Equation $3x^4 + 2x^3 + 9x^2 + 4x + 6 = 0$

Solve the equation $$3x^4 + 2x^3 + 9x^2 + 4x + 6 = 0$$ Having a complex root of modulus $1$. To get the solution, I tried to take a complex root $\sqrt{\frac{1}{2}} + i \sqrt{\frac{1}{2}}$ but ...
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79 views

Characterizing kernel of ring morphism

Let $K$ be a field and define a ring morphism $$\psi: K[x_1,x_2, \dots , x_n, y_1, y_2, \dots , y_n] \rightarrow K(x_1,x_2, \dots , x_n)$$ by $\psi(x_i) =x_i$ and $\psi(y_i) =\frac{1}{x_i}$. I ...
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Upper bound on the magnitude of the roots of a complex polynomial

Problem: Let $z_0$ be a root of the complex polynomial $z^n + a_{n-1}z^{n-1} + ... + a_0 $ $ (a_k \in \mathbb{C})$. Prove that $|z_0| \le \zeta$, where $\zeta$ is the only positive root of $z^n - ...
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75 views

If $p(z)$ is an injective polynomial $\Longrightarrow$ $p(z)=az+b$ [closed]

If $p(z)$ is an injective polynomial, how to prove that $p(z)=az+b$ with $a\neq 0$. $p(z)\in\mathbb{C}[z]$. Any hint would be appreciated.
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40 views

If $p(z)$ is a monic polynomial then $p(z)+b=(z-z_1)(z-z_2)\cdots (z-z_n)$

I need some help with this problem: If $p(z)$ is a monic polynomial of degree $n$ then there is a $b\in\mathbb{C}$ such that $p(z)+b=(z-z_1)(z-z_2)\cdots (z-z_n)$ where $z_1,z_2,\dots,z_n$ are simple ...
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1answer
41 views

Division by factorized polynomials in Macaulay2

I have this problem dividing by factorized polynomials, for example (x_1^4-x_2^4)//(factor(x_1^2-x_2^2)) does not work because the numerator is of "class R" (R is the ring kk[x_1..x_n]) and the ...
3
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1answer
54 views

Proper Field extensions

Given a field $F$, is there a proper field extension $K$ such that any root in $K$ of a polynomial in $F[X]$ is in $F$? Note: I am not looking for the algebraic closure of $F$. One candidate is the ...