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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16
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Number of monic irreducible polynomials of degree $p$ over finite fields

Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ can exist over $F$? Thanks!
6
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4answers
369 views

Problem: Sum of absolute values of polynomial roots

Can you please give me some hints as to how I might approach this problem? Thanks! Given the polynomial $f = 2X^3 - aX^2 - aX + 2, a \mathbb \in R$ and roots $x_1, x_2$ and $x_3,$ find $a$ such ...
10
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5answers
512 views

imaginary numbers - how can I understand them - especially as they occur in 'roots' of polynomials?

In another question here, about roots of equations being imaginary, the accepted answer said something interesting about "imaginary" (as a technical word in math) not meaning "not real". I ...
2
votes
2answers
364 views

Representing affine transform of Legendre polynomials

I have a function defined as a set of weighted Legendre polynomials: $f(x)=\alpha_0 P_0(x) + \alpha_1 P_1(x) + \alpha_2 P_2(x) +\ldots$. I have another function similarly defined with Legendre basis ...
4
votes
2answers
503 views

Integrate in Mathematica takes forever

I'm trying to calculate the length of a curve from a polynomial in Mathematica and for some reason Integrate doesn't complete, it just runs until I abort the execution. The polynom: ...
4
votes
3answers
300 views

Is there an algorithm to find the roots of high-order polynomials?

It is not generally possible to determine the roots of a polynomial whose grade is bigger than 4 in terms of roots and basic operations. But I heard, that it is possible to give a criteria whether a ...
3
votes
2answers
146 views

Euclids Algorithm for polynomials and a greatest common divisor

I have question about a problem I've encountered while attempting to solve an exercise (it's from an exercise in a homework series). Suppose we have two polynomials $f$ and $g$ (presumably over the ...
1
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0answers
117 views

Efficient way to recompute weights when shifting range of Legendre polynomial bases

I am storing a 2D (Cartesian) density function as a 2D patch with known X/Y limits and a set of 11 coefficients of the third order 2D Legendre polynomial basis functions over that patch. This works ...
2
votes
2answers
76 views

bound on the leading coefficient of a polynomial

Given a polynomial with real coefficients, that satisfies $\forall x \in \left[ -1,1 \right]: \ |p(x)|\leqslant 1$, I have to show, that its leading coefficient, $a_m$ satisfies $a_m \leqslant ...
9
votes
1answer
732 views

(Ir)reducibility criteria for homogeneous polynomials

Suppose I have a homogeneous polynomial in at least 3 variables over some algebraically closed field (of characteristic 0, if need be). Question: How may I test — by hand — whether it is irreducible? ...
5
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3answers
3k views

How to find roots of $X^5 - 1$?

How to find roots of $X^5 - 1$? (Or any polynomial of that form where $X$ has an odd power.)
2
votes
2answers
104 views

Bijection with constraints between sets of polynomial

How can I show the existence (or even better: construct explicitly) of a bijection (if one exists) between the set of all polynomial that have integer coefficients and a integer root and the set of ...
3
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1answer
1k views

$x^p -a$ is irreducible in a field of char $0$

Let $F$ be a field of characteristic $0$. How do I show that $x^p -a$ is irreducible in $F[x]$? Here $p$ is prime and $a$ is not a $p$-th power in $F$. This is a problem from Lang's ...
6
votes
4answers
176 views

Is there a usual method for finding the minimal polynomial of trigonometric values?

I've been thinking a bit about finding the minimal polynomials of side lengths of regular $n$-gons inscribed in the unit circle. For example, I recently wanted to find the minimal polynomial of the ...
3
votes
0answers
653 views

Existence of n distinct (real) roots of an orthogonal polynomial

I'm trying to get my head around the proof that an orthogonal polynomial ($P_n$ say) has at least n distinct roots. My understanding of the proof ...
3
votes
4answers
1k views

How to find a polynomial from a given root?

I was asked to find a polynomial with integer coefficients from a given root/solution. Lets say for example that the root is: $\sqrt{5} + \sqrt{7}$. How do I go about finding a polynomial that has ...
1
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2answers
215 views

Why is $R[X,Y]/(X^2-Y^3)$ isomorphic to $\{\sum a_iT^i\in R[T] \; : \; a_1=0\}$?

Let $R$ be a ring (commutative, with unit). Show that $A=\{\sum a_iT^i\in R[T] \; : \; a_1=0\}$ is a subring of $R[T]$ and isomorphic to $R[X][Y]/(X^2-Y^3)$. Of course, I'm trying to find a ring ...
1
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4answers
8k views

Factoring Methods/Tricks

One of the things I've struggled with most in algebra/calculus is all the "factoring tricks". When I take time away from doing math I inevitably forget most if not all of them. The old proverb "use it ...
4
votes
4answers
1k views

Finding double root. An easier way?

Given the polynomial $f = X^4 - 6X^3 + 13X^2 + aX + b$ you have to find the values of $a$ and $b$ such that $f$ has two double roots. I went about this by writing the polynomial as: $$f = X^4 - 6X^3 ...
5
votes
2answers
369 views

Is Gauss's lemma valid for polynomials with coefficients in a GCD domain?

Wikipedia's proof of Gauss's lemma requires this theorem: If $(C \mid S\cdot T) \land \lnot \operatorname{invertible}(C)$, $C$ has a non-invertible divisor in common with at least one of $S$ and ...
3
votes
2answers
107 views

Rational function sequence product

I want to know a closed formula for $$\prod_{m = 0}^{n} \frac{m^2 + a}{m^2 + a + 1},$$ a being any given complex. When the exponent is 1, it's pretty trivial, because of cancellations, but with other ...
4
votes
3answers
142 views

How far can we reach for the sum of roots in closed form for a polynomial of even degree?

As everybody knows, our reach for the roots themselves of a polynomial of any degree ends at degree 4, except in special cases. However, since the formula for the sum of the roots of a quadratic is ...
2
votes
0answers
129 views

Turning real roots into curves (for visualisation)

One can obviously map a set of real numbers $x_1, x_2, \ldots x_N$ to a curve in 2-D via $y=(x-x_1)(x-x_2)\ldots(x-x_N)$. Thinking about data visualisation, one can portray a set of $N$ observations ...
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1answer
2k views

asymptotically larger vs polynomially larger

What is the difference between x being asymptotically larger than y and x being polynomially larger than y?
3
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4answers
331 views

Factorize $x^3-3x+2$

How can I factorize $x^3-3x+2$ ? The answer that I got on the internet is $(x-1)^2(x+2)$. It would be nice if anyone could also tell what these type of equations are called and where can I learn ...
3
votes
2answers
128 views

Why must the constant of the divisor be negated in synthetic division?

Why must the constant of the divisor be negated in synthetic division? For example, if one was dividing a polynomial by $x+a$, $a$ must be changed to $-a$. Why is this?
7
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2answers
343 views

A Curious Binomial Sum Identity without Calculus of Finite Differences

Let $f$ be a polynomial of degree $m$ in $t$. The following curious identity holds for $n \geq m$, \begin{align} \binom{t}{n+1} \sum_{j = 0}^{n} (-1)^{j} \binom{n}{j} \frac{f(j)}{t - j} = (-1)^{n} ...
5
votes
5answers
589 views

A polynomial that is zero on an open set

Suppose that a polynomial $p(x,y)$ defined on $\mathbb{R}^2$ is identically zero on some open ball (in the Euclidean topology). How does one go about proving that this must be the zero polynomial?
5
votes
4answers
384 views

Show that $q(n)=11n^2 + 32n$ is a prime number for two integer values of $n$

Let $n$ be an integer and show that $q(n)=11n^2 + 32n$ is a prime number for two integer values of $n$, and is composite for all other integer values of $n$.
3
votes
1answer
133 views

Confused with logical jump in this algebra exercise

I can't seem to work out how my textbook got from the denominator($x^2 - x$) in the first line to the divisor below it ($x^2 +2x - 3$). Could anyone explain how they got from one to the other? ...
0
votes
1answer
111 views

Is there a name for the remainder of long polynomial division in sigma notation?

I was dividing polynomials and forgot to stop when the remainder's degree was lower than that of the dividend's. I had never done this before, so I continued and a very predictable sequence emerged. I ...
0
votes
3answers
116 views

How would I express $f(x)=2x^3-5x^2+3$ in the form $f(x)=(x-c)q(x)+r$ for $c = -2$?

How would I express $f(x)=2x^3-5x^2+3$ in the form $f(x)=(x-c)q(x)+r$ for $c = -2$? I have no idea where to even start. Could anyone help me out?
5
votes
0answers
75 views

Restriction of trivariate polynomial to $1$ variable

Let $p(x,y,z): \mathbb{F}^3 \to \mathbb{F}$ be a trivariate polynomial of degree $d \ll |\mathbb{F}|$. We choose uniformly at random an affine $1$-dimentional space $\ell = \{(a_1,a_2,a_3)t + ...
1
vote
2answers
121 views

Reverse search for rational function

Say we have two transcendental numbers, u and v. And u presumably can be obtained as a result of applying a rational function $Q$ with integer coefficients to v. Is it possible to find such rational ...
3
votes
1answer
169 views

Derivative of a homogeneous polynomial map

Let $K$ be a field and $V$ be a linear space over $K$. A map $p\colon V \to K$ is homogeneous polynomial of degree $n$ if there exist the symmetric $n$-linear form $f\colon V^{\times n}\to K$ such ...
7
votes
1answer
319 views

From complex solution to solutions over finite fields

There are several ways (Hilbert's Nullstellensatz, model theory, transcendence bases etc.) to prove the following (amazing!) result: If $f_1,...,f_r$ is a system of polynomials in $n$ variables with ...
4
votes
3answers
197 views

Two polynomial problems

I am struggling with these two problems: 1) Let $p$ be a polynomial with integer coefficients. Show that for each sequence of $k$-times iterating the polynomial, $n,p(n),p(p(n))=p^{(2)} ...
2
votes
1answer
67 views

Can not 'see' how to get next line of a particular Sturm Sequence

Stu(0)P = X^4 + pX^2 + qx + r Stu(1)P = 4x^3 + 2px + q Stu(2)P = -[2px^2 + 3qx + 4r]/4 Should anyone know how to get from the 3 lines above to Stu(3)P shown here on next line:- Stu^3(P) = ...
1
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0answers
147 views

Orthogonal polynomial interpolation of a function

I want to write down an arbitrary function $f$ as an (infinite) sum of orthogonal polynomials, e.g. for $f(x) = e^{\sin(x)}$, $f(x) = \sum{a_n T_n}$, where $a_n$ are the coefficients and $T_n$ are the ...
1
vote
2answers
2k views

problem to determine the chromatic polynomial of a graph

for a homework graph theory, I'm asked to determine the chromatic polynomial of the following graph this is my thread in another post: ...
3
votes
2answers
839 views

When polynomials f(x) and f'(x) are relatively prime, f(x) has no repeated roots. Why?

The problem is to show that a polynomial $f(x) \in F[x]$ (F is a field) has no repeated roots if and only if f(x) and f'(x) (the derivative of f(x)) are relatively prime. I've managed to prove one ...
11
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1answer
578 views

What's the name of a parabola mapped onto a sphere?

It seems that an 'arc' is a line-segment mapped onto the surface of a sphere (although I don't know if that name still holds if the segment wraps around the sphere more than once, i.e., if the angle ...
1
vote
3answers
911 views

How to determine the degree of a polynomial?

If $$g(x) = x^4 + x^3$$ From my understanding, the degree of the above polynomial i.e. $g(x)$ is 4. However, for this polynomial, $$f(x) = (x-1)(x-2) \cdots (x-p+1)$$ What degree does $f(x)$ have? ...
7
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1answer
534 views

How to solve polynomials?

Using Galois theory we can effectively compute whether or not a polynomial is solvable in radicals - technically this means you can build a chain of field extensions by adding $n$-th roots of ...
4
votes
1answer
215 views

What is the formula for this function $f(x) = (x-1)(x-2)(x-3) \cdots (x-k)$

I wonder if there exists a formula for this function? $$f(x) = (x-1)(x-2)(x-3) \cdots (x-k)$$ I want to know the coefficient of each $x^i$, and the first thing I came up with was to find the expansion ...
3
votes
2answers
116 views

The equation $F(x) \equiv 0 \pmod m$ has integer solution for x

Let $F(x)=(x^2-17)(x^2-19)(x^2-323)$ and let $m$ be a positive integer. How can one show that the equation $F(x) \equiv 0 \pmod m$ has an integer solution?
11
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1answer
495 views

Hermite's solution of the general quintic in terms of theta functions

Can someone point me at or produce a translation or modern exposition of Hermite's solution of the general quintic in terms of theta functions? (the "before" and "after" steps are on the mathworld ...
2
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1answer
113 views

Generating polynomials that are co-prime to their first and second derivatives

Let $f \in \mathbb Q [X]$ and not constant or of the form $(x-a)^n$. Suppose: $f_1 := \frac{f}{gcd(f,D^2f)}$ and; $f_2 := \frac{f_1}{gcd(f_1,Df_1)}$, where $Df$ stands for the formal derivative. ...
0
votes
1answer
585 views

Solving Polynomials in Computer Algebra Systems

Apart from low degree polynomials (2, 3, and 4) and factoring to lowest degrees, what are the method(s) used to find all the roots of a high-degree polynomial equations having only complex roots, and ...
1
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2answers
97 views

Finding a third degree equation that fits two points with given slopes

I'm trying to find an easy way of getting coefficients of a third degree polynomial $y = ax^3 + bx^2 + cx + d$ with given points $(x_1,y_1)$ and $(x_2,y_2)$ the slopes are also given $k_1$, $k_2$. I ...