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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Types of polynomial functions. Quadratic, cubic, quartic, quintic, …,?

I would very much like to have a complete list of the types of polynomial functions. I know that theres: ...
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139 views

Basis of a submodule of $\mathbb{Q}[X]^3$ over $\mathbb{Q}[X]$

I want to find the basis of a submodule of $\mathbb{Q}[X]^3$ generated by $(2X-1, X, X^2+3)$, $(X,X,X^2)$ and $(X+1,2X,2X^2-3)$. I determine that $$ \begin{pmatrix} 2X-1 & X & X^2+3 \\ X ...
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Interscholastic Mathematics League Senior B #12

Compute the product of the nonreal roots of the equation $x^4+4x^3+6x^2+1004x+1001=0$. So here is what I have done so far. I got two of the roots to be zero and 4 since ...
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1answer
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Interscholastic Mathematic League Senior B Division #11

The roots of the equation 3x^3-38x^2+cx-192=0 form a geometric progression. Compute c.
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Reed Solomon Polynomial Generator

I am developing a sample program to generate a 2D Barcode. And i am using reed solomon error correction code. By Going through this article i am developing the program. But i couldn't understand how ...
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Polynomial intersecting the exponential

Given a real polynomial $p(x)$ of degree $n$ such that $p$ and all its derivatives are nonnegative on some open interval $I$, how many times can $p$ intersect the exponential function (in $I$)? In ...
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1answer
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How do I factor this kind of equations?

I was doing some integration by partial fractions exercises and I found this equation:$$\int_{0}^{1}\frac{x^{3}+1}{x^{4}+4x+3},$$ and I don't know how to factor that in order to compute the partial ...
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Roots of $f_n(x) = 1 + (1-x)^2 - (x+3)(1-x)^{n+1}$ in the interval $[0,1]$

Does the polynomial $f_n(x) = 1 + (1-x)^2 - (x+3)(1-x)^{n+1}$ have exactly one root in the interval $[0,1]$ for all non-negative integers $n$? It has at least one root because $f_n(0) = -1$ and ...
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What's the easiest way to calculate a power series/polynomial modulo a prime?

I'm learning about polynomials stored in a closed form that resembles a generating function or power series. Generally speaking, I have fractions of polynomials, with one example being ...
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1answer
75 views

Derivative of polynomial function

$$f(x) = 2x - \frac{1}{4} x^2$$ How could I know calculate with limits the derivative of this function when $x_0 = -4$? I started already like this: $$f'(-4) = \lim_{x\to-4} ...
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1answer
78 views

Relation between a polynomial and its reflection

I have two polynomials: $Q(z)=q_0 +q_1 z + \cdots q_mz^m$ and its reflection $ Q^'(z)=q_0 z^m +q_1z^{m-1}+ \cdots q_m$. I'd like to find a relation between them (i.e. $Q(z)= \phi(Q'(z))$, so ...
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921 views

Isolate a variable in a polynomial function

How would I go about isolating $y$ in this function? I'm going crazy right now because I can't figure this out. The purpose of this is to allow me to derive $f(x)$ afterwards. $$ x = \frac{y^2}{4} + ...
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1answer
52 views

Translating problems into polynomial ones

I know the question will feel rather vague, but here it goes anyway. In some research I've seen, people often translate their problem into polynomial ones. The theory of linear feedback shift ...
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Intriguing polynomials coming from a combinatorial physics problem

For real $0<q<1$, integer $n >0 $ and integer $k\ge 0$, define $$[k, n]_q \equiv -\sum_{m=1}^{n} q^{m(k+1)} (q^{-n}; q)_m = -\sum_{m=1}^{n} q^{m(k+1)} \prod_{l=0}^{m-1} (1-q^{l-n})$$ ...
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1answer
277 views

Representation of Fourier Transform's vectors

I am just learning FFTs and I am trying to debug a problem in MATLAB. I think I don't understand how is MATLAB's FFT function handling the polynomial powers, or I am doing something wrong manually. ...
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What important problems require one to solve large systems of polynomial equations?

What is an extremely important problem that requires one to solve large systems of polynomial equations? I've heard of a number of "general areas" where the problems crop up (robotics, coding theory, ...
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Factoring multivariate polynomials

If I have a multivariate polynomial $P[X_1,\dots,X_n]\in \mathbb{R}[X_1,\dots, X_n]$, is there a polynomial time algorithm to factor the polynomial into irreducible polynomials $\in ...
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1answer
407 views

Why is this polynomial irreducible over $\mathbb{Z}[i]$?

There is a passage on the crazy project saying $x^3+12x^2+18x+6$ is irreducible over $\mathbb{Z}[i]$. I'm trying to use Eisenstien's Criterion to figure it out. I know that 3 is irreducible in ...
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694 views

Matching of polynomial coefficients

I am trying to find the proof/theorem that states: Given two polynomials in x, if they are equal to eachother, their coefficients must also be equal For example, in ax^3 + bx^2 + cx + d = ...
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367 views

Question about a recursively defined function

Problem. Let $(f_n)_{n=1}^\infty$ be a sequence of functions $f_n\colon [-1,\infty)^n\to\mathbb{R}$ that are recursively defined in the following way: $$f_1(x_1)=1+x_1,$$ $$f_n(x_1,\ldots,x_n) = ...
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1answer
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On a Corollary to Gauss' Lemma

Suppose $R$ is a UFD, with $F$ its field of fractions. A usual corollary to Gauss' Lemma on the content of polynomials states that if $f(X)\in F[X]$ has a factorization $f(X)=g(X)h(X)$ in $F[X]$, ...
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Two quadratic equations with equal ratios of roots

If the ratio of roots of $ax^2+bx+c = 0\space$and $px^2+qx+r = 0\space$is same. How to find ratio of their discriminants? I don't understand this problem,what exactly is meant by ratio of the ...
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Polynomial equations in 2 variables with symmetry

Suppose $P(x,y)$ is a polynomial with real coefficients. Is it true that any solution $(x_0,y_0)$ of the system $P(x,y)=P(y,x)=0$ has the property that $y_0 = \overline{x_0}$ (i.e. they are ...
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Low degree approximation of the polynomial extension of the logical-or function

Let $x\in\{0,1\}^n$ be a binary vector of dimension $n$, and let $OR(x)$ be the "logical or" function (i.e., returns $1$ if at least one of the coordinates is $1$ and otherwise returns $0$). Consider ...
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Extending the logical-or function to a low degree polynomial over a finite field

Let $x\in\{0,1\}^n$ be a binary vector of dimension $n$, and let $OR(x)$ be the "logical or" function (i.e., returns $1$ if at least one of the coordinates is $1$ and otherwise returns $0$). Is there ...
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1answer
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Leaving Cert Math Long Division

Solution to problem Hi, I'm correcting my work for study, and I cant get my head around this sum. I understand where the $x^2 + x − cx$ comes from but then when the 6 appears it loses me.
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A property of roots of the truncated series for $\sin(x)$

Let $p_n(x) = \sum\limits_{k=0}^n \frac{(-1)^kx^{2k+1}}{(2k+1)!}$ In other words, $p_n$ is the polynomial made of the first $n$ terms of the Taylor expansion of $\sin(x)$ around $x = 0$. ...
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1answer
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Basic question about polynomials

I understand polynomials in one variable as an algebraic expression that is made up of many terms, which consists of coefficients for example $a_n\dots a_0$ that are real numbers. $$a_n x^n+ a_{n-1} ...
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1answer
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Comparing algebraic varieties over a shared subset of variables

I'm currently experimenting with polynomial ideals and Gröbner bases, and I seem to be lacking some terminology/understanding. I have two systems of polynomial equations $P$ and $Q$ over a field ...
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Characteristic polynomials exhaust all monic polynomials?

Let $A$ be an $n\times n$ matrix, then $\mathrm{char}_A(x):=\det(xI-A)$ is a monic polynomial of degree $n$. It is called the characteristic polynomial of $A$. My question is the converse: Let ...
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Low degree extension

Let $v$ be a binary vector of dimension $n$. Assume that $n$ is a perfect square, then $v$ can be thought of as a function $f:[\sqrt n]\times[\sqrt n]\to \{0,1\}$, where $[\sqrt n]=\{1,\ldots,\sqrt ...
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1answer
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Expressing the number of non-zero rows of a binary matrix as a polynomial

Let $X$ be an $m\times n$ matrix, such that all of its elements are binary, i.e., for every $1\leq i\leq m$ and $1\leq j\leq n$ holds $x_{ij}=(X)_{ij}\in\{0,1\}$. Is there any possible way to express ...
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Irreducible polynomial of $\mathrm{GF}(2^{16})$

I'm implementing some code for the Galois field $\mathrm{GF}(2^{16})$. Which irreducible polynomial do you recommend that I use?
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What is meant by "All Polynomials of the form $p(t) = a + t^2$?

I have a math homework problem that goes like this: Determine if the given set is a subspace of $\mathbb{P}_n$ for the appropriate value of $n$: All polynomials of the form $p(t) = a + t^2$, where ...
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Name of property describing the number of times a function changes concavity?

For example, $f(x)=\sin x$ changes concavity an infinite number of times, $f(x)=x^3-x$ has two regions of concavity (changing concavity once), and $f(x)=x$ changes $0$ times. Is there a name for ...
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Finding the real root of a cubic algebraically

I'm sorry if this is a very easy question but my brain is fried tonight and I can't think how to do it. I need to solve $x^3 = 2 - x$. Obviously by eyeballing the equation you can see that the only ...
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1answer
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Roots of bivariate polynomials

A bivariate polynomial of degree $m+n$ is, $ p(x,y) = \sum_{k=1}^n\sum_{j=1}^m a_{jk}x^ky^j$ where $a_{mn}\neq0$ and $a_{jk}\in\mathbb{R}$ for $1\leq j\leq m$, $1\leq k\leq n$. Univariate ...
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Linear Algebra Question (Polynomial Interpolation)

Given the data for an experiment: Velocity: 0, 2, 4, 6, 8, 10 Force: 0 , 2.9, 14.8, 39.6, 74.3, 119 (One force value listed below one velocity value in a table) Find an interpolating ...
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1answer
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Variation over univariate Schwartz–Zippel lemma

Let $n\in\mathbb{N}$ and let $q\in[n,2n]$ be a prime number. In addition, let $s,s':\mathbb{F}_q\to\mathbb{F}_q$ be polynomials of degree $\sqrt{n}$ such that $s\neq s'$. From the ...
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1answer
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orthonormal polynomials

Here is the question: Suppose $P_0, P_1, P_2, \dots$ are polynomials orthonormal with respect to the inner product $$(f,g)=\int_a^b f(x)g(x)W(x)dx,$$ where $W(x) > 0$ is a weight function and ...
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Create polynomial coefficients from its roots

Given some roots : $r_1,r_2,\ldots,r_n$, how can we reconstruct polynomial coefficients? I know the Horner scheme and that we can just go backwards receiving those coefficients. But I'm curious if ...
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What is the simplest ellipse that goes through exactly 13 lattice points?

The ellipse $-30 x + 3 x^2 - 10 y - 3 x y + 4 y^2$ goes through exactly 11 lattice points. Another such ellipse is $4 - 30 x + 2 x^2 - 5 y - x y + 3 y^2$. What is the simplest ellipse that goes ...
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Is there a neat way to show $\int_{-1}^1 \frac{ U_n(z) U_n(z)}{\sqrt{1-z^2}} \mathrm{d} z = \pi (n+1)$

In answering a question on math.SE, I attempted to find integral of Fejér kernel by using $$ K_n(t) = \frac{1}{n} U_{n-1}^2\left( \cos \frac{t}{2} \right) $$ where $U_n(z)$ stands for the ...
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Proving that a polynomial is irreducible over a field

What's the general strategy to show that a particular polynomial is irreducible over a field? For example, how can I show $x^4 - 10x^2 -19$ is irreducible over $\mathbb Q$?
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1answer
187 views

inverse polynomial

I am reading from some notes on cryptography and came across this sentence: "We call a function f negligible in k if it asymptotically approaches zero faster than any inverse polynomial in k i.e., ...
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Finding roots of polynomials with rational coefficients

I'm looking for a general approach (or approaches) for finding the roots of polynomials with rational coefficients of higher degrees than $4$. The problem is that I need to find the exact roots and ...
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Big O of polynomial functions

I am required to identify if $\log{(f(x))}$ is a subset of $O(\log{n})$ holds true for all polynomial functions. If I try with $f(x) = x^2$, then I am able to prove it to be correct. But, with $f(x) = ...
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Polynomial Factorization

I was handed $x^3-x^2+x-2=0$ to factor, but I'm not sure how. I tried all the methods I know of--which, at the time of writing, are limited by my precalc math background (I'm working on that...). Is ...
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Partial fraction expansion when the degree of the numerator is unknown

Hope it's not too stupid: is there any general approach to partial fraction expansion when the degrees of polynomials in the numerator are unknown?
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1answer
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Solution to polynomial $ax^k-bx^{k-1}+b-a=0$

I once spent far too long getting nowhere with this. Is there a way of finding the real roots of $ax^k-bx^{k-1}+b-a=0$ where $a, b, k\in \mathbb N$ and $b\gt a$ and $k\gt 1$? I know that there is no ...