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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Discriminant function for general polynomials

According to Wikipedia... (terrible intro) The discriminant of a 6-degree polynomial has 246 terms. The article claims that the relationship between the terms in the discriminant has an exponential ...
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Power (monomial) form conversion to Chebyshev form

Given a polynomial in the monomial form e.g. like $p(x) = a_0 + a_1 x + \ldots + a_{n-1} x^{n-1} + a_n x^n$, how is it possible to convert it to the Chebyshev basis (i.e. represent it as a linear ...
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Polynomial division/deflation with FFT

There is a need to divide a polynomial $p(x)$ by polynomial $q(x)$, whereas it is known that the remainder will be zero (i.e. the question is about polynomial deflation). A known method is to use the ...
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PRIMES is in P, page 4: Why is $(X+a)^{\frac{n}{p}} \equiv X^{\frac{n}{p}}+a$ implied?

PRIMES is in P, page 4, equasion (5) Edit: I should probably add that $p$ is a prime factor of some $n$. $a$ is any number from 1 to some irrelevant limit. $r$ also shouldn't matter because as far as ...
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Contradiction - Equivalence of polynomials

I think I'm having a brain fart. Please tell me if my reasoning is correct. Suppose you have a polynomial-function $f(x)$ of degree $N$ that has coefficients $a_{0 \leq j \leq N}$ and roots $r_{0 ...
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List primitive elements of GF(2^3) = {0, 1, a, a^2,…, a^6} [on hold]

I need the find the primitive elements of GF(2^3) = {0, 1, a, a^2,....., a^6}, could any one help me out how to go about it?
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How many primitive elements does GF(256) have?

I know the answer for this is 36 but I don't exactly know how to reach to this. Can you any one help me in understanding this.
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Polynomial Long Division with Divisor<Dividend

So here's the problem... 20x^3-4/5x^2-3 When I divide this I get 20x^3 -4 -20x^3 +12x 12x-4/5x^2-3 So 5x^2 goes into 12x how many times? It doesn't seem to. So how do I solve this?
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Ladder against a wall.

Having a bit of a problem with a question. There is a 4m ladder leaving against a wall. There is a box in between The ladder and wall. The box is a cubic metre. I have found a quartic to find the ...
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1answer
23 views

Existence of a root $\alpha$ so that $|\alpha+i| <1$

For some monic polynomial $P(z) = \displaystyle \sum_{k=0}^n a_k z^k, 0 < |P(i)| < 1, a_k \in \mathbb{R}, k=0,1,...,n$, how does one show that a complex root $\alpha$ exists such that $|\alpha + ...
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Fully factorise $x^3-x^2-14x+24$ into linear factors

$$f(x)=x^3-x^2-14x+24$$ I've tried grouping the terms, but it just doesn't work out for me. Any help is appreciated.
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Methods for determining which roots of a polynomial are inside of the unit circle?

Let's say I have a polynomial such as $$p(x) = x^4 + bx^3 + cx^2 + bx + 1.$$ I strongly suspect that, for any parameters, there are always two roots inside the unit circle and two roots outside of ...
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1answer
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Can the induced function of non-zero $f \in R[X]$ be zero, when $R$ is an infinite non-integral domain?

Let $R$ be an infinite commutative ring with $1$ which is not an integral domain. Is it possible to have a non-zero $f\in R[X]$ such that the induced map $\bar{f}: R \to R$ is zero? Please give ...
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Presentation of a module by generators and relations

Let $R:=\mathbb C[T]$. Match the $R$-module with the presentation by generators and relations. $\bullet$$R$-modules: $M:=\mathbb C[T,T^{-1}]$ (Laurent Polynomials)$\qquad$$N:=\mathbb ...
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Linear functionals and integration verification

Can you please verify my reasoning? (a) Yes as (b) No, as function is squared (c) Yes, same reasoning as (a), squared values of x do not affect linearity. Does the region of integration affect ...
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1answer
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What is a quick proof that $f \in \mathbb{C}[X_1,\dotsc,X_n]$ is determined by its induced function on $\mathbb{C}^n$?

For $f \in \mathbb{C}[X_1, \dotsc, X_n]$, we have the induced function $\bar{f}: \mathbb{C}^n \to \mathbb{C}$ given by evaluation. The association $f \mapsto \bar{f}$ is injective. Is there a quick ...
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Is this closed form of $\int_0^1 \operatorname{Li}_3^2(x)\,dx$ correct?

According to Freitas' paper at page $11$. $$\int_0^1 \operatorname{Li}_3^2(x)\,dx = 20-8\zeta(2)-10\zeta(3)-\frac{15}{2}\zeta(4)-2\zeta(2)\zeta(3)+\zeta^2(3).$$ I evaluated the LHS and it is ...
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5answers
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How to find the complex solution of $x^6$

How do you find the complex solutions to $x^6+x^3-2=0 $ Obviously $x=1$ is one solution, but i cant get further than that.
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On even cyclotomic polynomials

Let $\Phi_n$ be the nth cyclotomic polynomial. I would like to show that if $4$ divides $n$, then $\Phi_n$ is even. Any idea ?
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yacas factorize polynoms

I want factorize polynoms with yacas but I can do it only with univarial. E.g. I want $x^2-y^2$ factorize to $(x-y)(x+y)$. How can I do it? Or if anybody has any suggestion to another simple, free ...
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+50

show that this polynomial can't multiple root occurring more $n-1$ times

Question: let $x_{1},x_{2},\cdots,x_{n}$ be a complex numbers,and such $x_{i}\neq x_{j},\forall i\neq j$, show that: following this polynomial can't $$p(x)=(x-x_{1})^2(x-x_{2})^2\cdots ...
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Alexander-Conway polynomial of an unlinked knot…

I had asked this elsewhere earlier in the week but I decided I am more likely to get an answer here: Is it true that for all unlinks, the Alexander-Conway polynomial is equivalent to 0? It seems ...
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Is solving the quintic the obstacle to solving the Riemann hypothesis?

Mathematica knows how to solve: ...
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Roots of product of two polynomials is the union of the roots of each polynomial

I'm trying to prove this lemma: The roots of $P(x)*Q(x)$ is the union of the roots of $P(x)$ and $Q(x)$ for all $x$. It's trivially true, which is why I find it hard to prove. Let $r(x) = ...
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Number of distinct real roots of $x^9 + x^7 + x^5 + x^3 + x + 1$

The number of distinct real roots of this equation $$x^9 + x^7 + x^5 + x^3 + x + 1 =0$$ is Descarte rule of signs doesnt seems to work here as answer is not consistent . in general i would like to ...
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Closed form of $\int_0^1 B_n(x)\psi(x+1)\,dx$

Is there a closed-form of the following integral? $$I_n = \int_0^1 B_n(x)\psi(x+1)\,dx,$$ where $B_n(x)$ are the Bernoulli polynomials and $\psi(x)$ is the digamma function. The motivation of the ...
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How would I determine a single equation for a set of points

Four different functions are bounded by certain values along the $x$-axis. What I want to know is if there is one function that can describe all points in the set. To be more specific, I have four ...
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How to find the 4th degree polynomial with given values at $0,1,2,3,4$?

Determine a fourth degree polynomial p that has $p(0), p(1), p(2), p(3), p(4)$ equal to $7, 1, 3, 1, 7$, respectively. Using my ideas, I first write out the points on the polynomial as $(0,7), (1, ...
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How could we define a sheaf or presheaf of polynomials? [on hold]

Good evening everyone , Is there a sheaf or presheaf whose sections are polynomials defined on opens of a topology ? . If yes , what is this topology ?. Is it the Zariski topology , and why? And how ...
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What is the quickest way to find the characteristic polynomial of this matrix?

Let $e_k$ be the $k$-th vector of the canonical base of $\mathbb R^n$ and let $$B = [e_2 \mid e_3 \mid \dots \mid e_n \mid e_1]$$ What it the quickest way to show that the charachteristic polynomial ...
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Obtain Base in this equation

If we have below equation and know that $6$ and $3$ are answers of this equation obtain base of these equation: $$X^2 - 11X + 22 = 0$$
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Polynomial approximation

Say that you have $n+1$ points on the interval $[a,b]$, let's call them $\{x_0,\dots,x_n\}$. Take any two different $y_1, y_2$, points on $[a,b]$. My goal is to show that there exists a polynomial $p$ ...
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Fun with Newton's Method - Infinitely many cycles

I'd like to preface this problem by saying that I have absolutely no clue if it is solvable or not. This is just the result of some musings, and I'm looking for either some guidance, or to be pointed ...
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Showing that an ideal is maximal

Let $k$ be an algebraically closed field and $f$ be the polynomial $x_1x_2+x_2x_3+x_3x_1$ in $k[x_1, x_2, x_3]$. Here $f$ is irreducible. Then this polynomial ring is not a $PID$, it is only an ...
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Completely factor a polynomial using the rational root theorem and synthetic division

I am currently seriously confused. My problem, as stated above, is about completely factoring a polynomial. My question is, once you get your possible factors, how do you then simplify it down? Ill ...
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Two equations have the same number of roots. [on hold]

Find functions $f(x)$ such that $\forall\,(a;\,b)\ne (0;\,0)$ two equations $f(x)=ax+b$ and $x^2=ax+b$ have the same number of roots (real roots).
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Difference table for interpolation

For calculating divided (fraction) difference table for interpolating the points $(x_i, f_i)$, $i=1,2,...,n$; by using a polynomial with degree lower or equal to $n$, $n(n-1)/2$ fraction was used. I ...
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Show that a polynomial $P(x)$ has $r$ as a double root if and only if $P'(r)=0$ and $P(r)=0$

Assuming that $r$ is a double root. Then $$P(x)=(x-r)^2\cdot k(x).$$ We also have the derivative: $$P'(x) = 2(x-r)k(x) + (x-r)^2k'(x).$$ Hence, $$P(r) = (r-r^2)k(r)=0$$ and $$P'(r) = 2(r-r)k(r) + ...
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polynomial over finite field, roots forming additive subgroup

Let $q=2^h$ and $t=2^r$ for some $h\ge r$ and denote by $\mathbb{F}_q$ the finite field of order $q$. (since the previous, simple version was wrong, I'm posting here a new version) Let $f$ be a ...
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Find gcd and lcm of two polynomials

Let $f(x)=x^3+x^2+x+1$ and $g(x)=x^3+1$. Then in $\mathbb{Q}[x]$ $\gcd (f(x),g(x))=x+1$ $ \gcd(f(x),g(x))=x^3-1$ $\operatorname{lcm}(f(x),g(x))=x^5+x^3+x^2+1$ ...
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Matlab Code about evaluating Newton Polynomials

I am trying to write a code for evaluating a newton polynomial with coefficients $a = [a_0 , ... , a_n]$, and nodes $x = [x_0 , ... , x_n]$ at the vector $t$, using nested multiplication. ...
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Linear functionals and dual bases

How do I tackle this question? I am a little hazy on linear functionals and integral signs.
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Polynomial decomposition

I've just recently learned about the neat algorithm that, given a polynomial $f$ finds (non linear) polynomials $h,g$ such that $$f = g \circ h \quad (1),$$ or decides that there are no such ...
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Herstein - Topics in Algebra - Polynomial rings page 157

In Chapter 3.9 of his book "Topics in Algebra" , 2nd ed, Herstein describes an example of a Quotient ring, namely $ F[x]/(x^3-2) = F[x]/A $ where $F = Q $ the rationals, and $(x^3-2) = A $ is the ...
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Simplifying the difference quotient $\frac{(x + h)^3 - x^3}{h}$.

For the function $f(x) = x^3$, I have the difference quotient: $$ \frac{(x + h)^3 - x^3}{h} $$ I tried changing the $(x + h)^3$ to $(x + h)(x^2 - xh + h^2)$ that I know to see if I could get ...
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Show some polynomial is irreducible over the field of 7 elements.

I have to show that the polynomial $x^4+x^3+x^2+x+1$ is irreducible over the field $F_7$. It doesn't have roots in $F_7$, but I can't show it does not have degree two irreducible factors in $F_7[x]$. ...
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Factorize $6x^2 -5x -14 = 0$

I'm throwing a bit of a blank on the best way to factor this : $$6x^2 -5x -14 = 0$$ I know that I could multiply $6$ by $14$ and then find a pair of factors that add to $-5$ (b), but this feels a ...
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+100

Simplifying polynomials

Suppose I have a (multivariate) polynomial with coefficients in $\mathbb Z$ or $\mathbb Q$, given in fully expanded form. How can I simplify this to reduce the number of elementary operations ...
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Properties of Roots of polynomials

Today in highschool we were doing a chapter called "Roots of polynomials" where we learnt something new and interesting which is : $ax^2+bx+c=0$ Has roots $\alpha$ , $\beta$ Then: $$\alpha + ...
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Space of real polynomials of one variable isn't complete

Consider $E$ - the vector space of all real polynomials of one variable. I need to prove that it is not complete under any norm. I was thinking I could use the fact that certain functions, for ...