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.

learn more… | top users | synonyms

1
vote
0answers
27 views

Field of polynomials mod n?

I have a few questions and i am looking for some clarification. 1) Is it correct that one can define a field $(Z_n, +, X)$ of integers mod $n$, where all the elements are integers $a$ such that ...
10
votes
2answers
142 views

Show that the roots of the polynomial $x^4 - px^3 + qx^2 - pqx + 1 = 0$ satisfy a certain relationship

Here is the question: If the roots of the equation $$ x^4 - px^3 + qx^2 - pqx + 1 = 0 $$ are $\alpha, \beta, \gamma,$ and $\delta$, show that $$ (\alpha + \beta + \gamma)(\alpha + \beta + ...
1
vote
0answers
30 views

Affine Hecke algebras and Lusztig relations

I study the book "Affine Hecke algebras and orthogonal polynomials" by I.G. Macdonald. He propose a formula in section $4.2$, especially formula $(4.2.9)$. This formula is the following: ...
2
votes
0answers
25 views

Show that $\Phi_{p^n}(X)=\Phi_p(X^{p^{n-1}})$ with $p$ prime. [duplicate]

How to show that $\Phi_{p^n}(X)=\Phi_p(X^{p^{n-1}})$ with $p$ prime where $\Phi_n(X)$ is the cyclotomic polynomial define by ...
1
vote
1answer
44 views

A big contradiction in interpolating point and number of it's

For calculating divided (fraction) difference table for interpolating $(x_i, f_i)$, $i=1,2,...,n$; by using a polynomial with degree lower or equal to $n$, $n(n+1)/2$ difference fraction was used. I ...
1
vote
1answer
58 views

Multivariate polynomials at bounded evens

Univariate polynomials Given $n$, is there a degree $cn^{c'}$ polynomial $p(x)\in\Bbb R[x]$ and a degree $dn^{d'}$ polynomial $q(x)\in\Bbb R[x]$ with fixed $c,c',d,d'>0$ such that $$m\in\Bbb ...
4
votes
1answer
51 views

Why does the Bezier Curve work?

Recently I've been looking at Bezier curves and trying to understand how they work. I know that a general Bezier curve is given by the equation $$ \vec{\mathbf{B}}(t) = \sum_{k=0}^n{b_{k,\ ...
20
votes
1answer
164 views

Solve $x^4+3x^3+6x+4=0$… easier way?

So I was playing around with solving polynomials last night and realized that I had no idea how to solve a polynomial with no rational roots, such as $$x^4+3x^3+6x+4=0$$ Using the rational roots test, ...
6
votes
3answers
635 views

Does this polynomial exist?

I'm looking for a polynomial $P(x)$ with the following properties: $P(0) = 0$. $P\left(\frac13\right) = 1$ $P\left(\frac23\right) = 0$ $P'\left(\frac13\right) = 0$ $P'\left(\frac23\right) = 0$ ...
1
vote
2answers
51 views

Babylonian solution to general cubic

Problem: Otto Neugebauer believes that the Babylonians were quite capable of reducing the general cubic equation to the "normal form" $n^3 + n^2 = c$, although there is as yet no evidence that they ...
2
votes
1answer
57 views

$P(x)\mid\sum_{i=0}^n a_ix^{2^i} $

Let $n \in \mathbb{N}^*$, and $P(x) \in \mathbb{R}[x], \deg P=n$. Prove there exist $a_0,a_1,...,a_n$ such that $a_0^2+a_1^2+\cdots+a_n^2 \not=0$ and $P(x)\mid \sum_{i=0}^n a_ix^{2^i}$.
3
votes
1answer
80 views

Show that the equation $x^{4} + rx + s = 0$ has at most two distinct real roots.

Show that the equation $x^{4} + rx + s = 0\:$ has at most two distinct real roots. Also, find a condition on $r$ and $s$ which ensures there are two distinct roots. This seems like it should be easy, ...
2
votes
1answer
37 views

Formal interpolation derivation of polynomial

I have a polynomial $$F(x_1,x_2,x_3,x_4)=k(x_1+x_2)(x_3+x_4)$$ The polynomial can be described by $$F(x_1,x_2,x_3,x_4)=0\iff (x_1+x_2)=0\mbox{ or }(x_3+x_4)=0$$ Is there a way to formally derive ...
1
vote
2answers
52 views

Polynomial whose roots are also the coefficents

Let the roots of a polynomial be $a_0<a_1<a_2<\ldots<a_{n-1}$, all integer and distinct. Suppose the polynomial can also be expressed as $x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$. Find all such ...
3
votes
1answer
116 views

Combinatoric Coefficients of a Polynomial

I have the following function: $$f(x)=\left(T_{N_2}(x)-T_{N_1}(x)\right)\left(T_{N_3}(x)-T_{N_1}(x)\right)\left(T_{N_3}(x)-T_{N_2}(x)\right)$$ where $T_{N}(x)=1+x+\frac{x^2}{2!}+...+\frac{x^N}{N!}$ ...
1
vote
3answers
49 views

Factorizing a polynomial of degree 4 that has complex roots

While working on differential equations with constant coefficients I came across the following auxiliary equation: $r^4 - 4r^3 + 9r^2 - 10r + 6 = 0$. Initially I tried the hit and trial method for ...
-2
votes
2answers
84 views

Polynomial division $p(x) = x^4-(2m + 4)x^2 + (m-2)^2$ [duplicate]

For which values of $m$ can the polynomial $p(x) = x^4-(2m + 4)x^2 + (m -2)^2$ be factored into two non-constant polynomials whose coefficients are integers?
2
votes
0answers
31 views

Sum of first $n$ integer values of a polynomial as a polynomial in $n$

Let $p$ be a polynomial. Let $$S_p(n)=\sum_{k=1}^n p(k)$$ be the sum of the polynomial's values at the first $n$ integers. Splitting the sum along each term's monomials, pulling the common ...
1
vote
1answer
135 views

How to find algebraic connections between zeros of a polynomial?

Let $f(x)$ be an irreducible integer polynomial of degree $k$. Let $x_1,x_2,...,x_j$ be some zeros of $f(x)=0$ where $j<k$. How do I find identities of type $P(x_1,x_2,...,x_j) = 0$ where $P$ is ...
3
votes
0answers
58 views

Finding the sum of the coefficient squared of a polynomial

Given a polynomial $$f(x) = \sum_{k = 0}^n a_kx^n = a_0+a_1x+ \ldots + a_{n-1}x^{n-1}+a_nx^n $$ is it possible to find a general expression for $$ \sum_{k = 0}^n a_k^2 ?$$ For example, $\sum_{k = ...
6
votes
2answers
101 views

Degree Polynomials and Zeroes

"Find a degree $3$ polynomial that has zeros $-3, 4$ and $8$ and in which the coefficient of $x^2$ is $-18$." I've been trying to solve this problem, but I keep getting it wrong. I've worked with ...
5
votes
0answers
44 views

Does this simple problem using Vieta's formulas have deeper connections to elliptic curves?

A friend posed the following question to me: Suppose $p(x)=x^3+ax+b$ has one real root, $x_1$, and two non-real roots, $x_2$ and $x_3$. Compute $x_1$ in terms of $x_2$. By Vieta's formulas, ...
2
votes
1answer
34 views

Inner Product on polynomials over field of complex numbers

I am playing with the simplest of polynomial vector spaces - the Legendre polynomials (I hope I have that name right! :-) where $\langle P,Q\rangle = \int_{-1}^{+1}P(x)Q(x)dx$ This is straightforward ...
3
votes
0answers
58 views

Problem on symmetric polynomials

The following problem is from "Analysis I" by Amann/Escher. Exercise: There are obvious operations of $S_m$ on $\mathbb{N}^m$ and on $R[X_1,\dots,X_m]$. A polynomial $p\in R[X_1,\dots,X_m]$ is called ...
1
vote
1answer
34 views

$\ g(\sqrt{a^3}) \cdot g(\frac{1}{{a}^3}) \ge 1$ or $\ g(\sqrt{a^3}) \cdot g(\frac{1}{\sqrt{{a}^3}}) \ge 1$?

Let $ n\in\mathbb{N} $ and $ x_0,x_1,.....,x_n $ so that $x_0 + x_1 + .... x_n =1 $, $ a\in\mathbb{R} $ , $a>0$ and $\ g(t)=x_0t^n+x_1t^{n-1}+.....+x_n ,$ for all t $\in\mathbb{R} $. Is any of ...
1
vote
1answer
31 views

Can someone check my answer to this question about approximating a function with a polynomial

Let $f\in C([0,1],\mathbb{R})$. The n-th moment of f is defined to be $M_n=\int_0^1f(x)x^n\;dx,\;n\geq0.$ Show that if $M_n=0$, for all $n\geq 0$, then $f=0$. My answer; By the Weierstrass ...
1
vote
3answers
66 views

Polynomials in one variable with infinitely many roots.

Can a non-zero polynomial in one variable have infinitely many roots ? Can a non-zero polynomial in one variable have uncountably many roots ? Motivation : over $\mathbb Z/12\mathbb Z$, ...
3
votes
0answers
26 views

Multivariate polynomials with prescribed zeros

Let $P_k\subsetneq\Bbb R[x_1,\dots,x_n]$ be the set of degree $k$ multivariate real polynomials. Pick a subset $S$ of $\{-1,+1\}^n$ of size $|S|<\sum_{i=0}^k\binom{n}{i}$. We seek a polynomial ...
3
votes
3answers
82 views

Synthetic division by $ax^2+bx+c$.

I know that synthetic division can be used in order to find quotient $q(x)$ and remainder $r(x)$ of a polynomial $p(x)$ when it is divided by some linear polynomial like $x-c$. Now, does exist some ...
1
vote
3answers
84 views

Computing $\lim_{x \to \infty} \frac{|x^n|}{e^x}$

How to prove that $e^x$ goes faster to infinity than any polynomial of $x$ without using the Taylor expansion of $e^x$ or L'hopital rule? in other words, the proof that: $$\lim_{x \to \infty} ...
1
vote
0answers
77 views

SOS relaxations for polynomial optimization

I do not understand how SOS (Sum-Of-Squares) relaxation for polynomial optimization works in some cases. For instance, consider the polynomial optimization problem: \begin{equation} ...
1
vote
1answer
17 views

The sequence of functions $u_{n+1}(t)=u_n(t)+\frac{1}{2}(t-u_n^2(t))$ approximates $\sqrt{t}$ from below

The square root function on $[0,1]$ is approximated by a sequence of functions (polynomials) defined on $[0,1]$. The induction hypothesis is that you have functions such that $$0 = u_0\leq u_1\leq ...
4
votes
2answers
93 views

Find the number of polynomial zeros of $z^4-7z^3-2z^2+z-3=0$.

Find the number of solutions of $$z^4-7z^3-2z^2+z-3=0$$ inside the unit disc. The Rouche theorem fails obviously. Is there any other method that can help? I have known the answer by Matlab, but ...
5
votes
0answers
43 views

Show that $(1+a_1x+\ldots+a_rx^r)^k=1+x+x^{r+1}q(x)$

Fixed $k\ge 1$. Show that for each $r$, you can find $a_1,\cdot\cdot\cdot,a_r\in \mathbb{F}$ such that :$$(1+a_1x+\cdot\cdot\cdot+a_rx^r)^k=1+x+x^{r+1}q(x)$$ where $q(x)$ is a polynomial. Any ...
1
vote
2answers
48 views

An equation over $\Bbb F_{3^k}$

Does the equation $$x^2=2=-1$$ have solutions in any extension field of $\Bbb F_3$?
1
vote
2answers
47 views

Polynomial annihilator method $y''+4y=\sin^2(2x)$

The question asks to solve the equation by this method. I know how to annihilate $\sin(2x)$ by $(D^2+4)$ however i don't know for the case $\sin^2(2x)$. Thanks!
3
votes
0answers
82 views

Is there any 100% sure numerical method to find all roots in a polynomial equation of degree n without fail?

Is there any 100% sure numerical method to find all roots in a polynomial equation of degree n without fail? I do not find any method which solve polynomial equation without fail.
2
votes
1answer
20 views

Clarification on how to prove polynomial representations exist for infinite series

With reference to this question, I would like a clarification of the comment given by @Ant (but someone else could answer instead). I basically have 2 questions: Is there any formal way to prove ...
1
vote
4answers
40 views

Given that$(3x-1)^7=a_7x^7+a_6x^6+a_5x^5+…+a_1x+a_0$, find $a_7+a_6+a_5+a_4+…+a_1+a_0$

Given that$$(3x-1)^7=a_7x^7+a_6x^6+a_5x^5+...+a_1x+a_0$$find $$a_7+a_6+a_5+a_4+...+a_1+a_0$$ Is there anyway to do the question without using binomial theorem and expanding the expression on the ...
1
vote
3answers
63 views

Then,what is the value of $P(0) + P(4)$?

A polynomial $P(x)$ with leading coefficient 1 of degree 4 is such that $P(\alpha)= 0$ and its roots are $1, 2$ and $3$. Then,what is the value of $P(0) + P(4)$?
1
vote
1answer
34 views

Define a matrix power by some scalars

Suppose I have an nxn matrix, for example: $$A=\begin{pmatrix}6&-2\\8&-2\end{pmatrix}$$ How is it possible to define the matrix $A^9$ using two scalars $b,c$ in R s.t.: $A^9 = bA + cI$ I ...
1
vote
2answers
78 views

What is the value of $P(6)$? [closed]

A polynomial of degree $5$ with leading coefficient $2$ is such that $P(1) = 1, P(2) = 4, P(3) = 9, P(4) = 16, P(5) = 25$ . Then, what is the value of $P(6)$ ?
2
votes
1answer
62 views

Determine the center of ring of differential operators with coefficients in $\mathbb{C}[z_1,z_2]$

My goal is to determine what is the center of a ring $R$ generated by differential operators $z_i \frac{\partial}{\partial z_j}$ for $i,j \in \{1,2\}$ with coefficients in polynomial ring ...
0
votes
2answers
63 views

Solve $p_4(x) = x^4 −(2m + 4)x^2 + (m−2)^2 $such that $p_4$ is a product of two non-constant integer-coeficient polynomials

I'm having trouble getting the starting idea for a problem I've been presented with: I need to find values for m (integer) such that the following polynomial $p_4(x) = x^4 −(2m + 4)x^2 + ...
7
votes
2answers
129 views

Problem Solving Question With Polynomials

For any polynomial $p$ with real coefficients, let $$ S(p):= \{x\in \mathbb{R} \mid p(x) \in \mathbb{Z}\} $$ Prove that if $p$, $q$ are two polynomials such that $S(p) = S(q)$, then either ...
36
votes
5answers
480 views

If $f(x)=x^2-x-1$ and $f^n(x)=f(f(\cdots f(x)\cdots))$, find all $x$ for which $f^{3n}(x)$ converges.

Let $f:\mathbb{R}\to\mathbb{R}$ be the polynomial defined by $$f(x)=x^2-x-1$$ and let $$g_0(x)=f(x),\quad g_1(x)=f(f(x)),\quad\ldots\quad g_n(x)=f(f(f(\cdots f(x)\cdots)))$$ The positive root of ...
0
votes
2answers
69 views

Why does an $n$th degree polynomial have at most $n-1$ turning points?

How can one explain that polynomial of degree $n$ can have up to $n-1$ turning points and $n$ intersections with the $x$-axis? If it is easier to explain, why can't a cubic function have three or ...
0
votes
1answer
51 views

Sum of roots: Vietas formulas

The equation $x^4-x^3-1=0$ has roots $\alpha, \beta, \gamma, \delta$. Find the equations with roots $\alpha^6, \beta^6, \gamma^6, \delta^6$. I was able to do this using the substitution $y=x^3$. I ...
2
votes
0answers
74 views

Solving systems of polynomials with an oracle

I need to solve a system of polynomials. Let the variables be $x_1, \dots, x_n$, and let the polynomials be $f_1, \dots, f_n$ Let's say we have these conditions we can already assume: there are ...
0
votes
2answers
36 views

Express the polynomial $ax^2+2hxy+2gx+2fy+by^2+c$ in matrix notation

I'm given $$\begin{bmatrix}x & y & 1\end{bmatrix}*M*\begin{bmatrix}x \\ y \\ 1\end{bmatrix}$$ where $M$ is the polynomial $ax^2+by^2+2hxy+2gx+2fy+c$ in matrix notation. Im totally stumped ...