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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Proving an equation is a fuction

Prove that the equation $y^3 + 3xy -5x^3 + 1 = 0$ defines $y$ as a function of $x$ for all $x$ in the real numbers.
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1answer
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How to find $α^2(β^4 +γ^4 +δ^4)+β^2(γ^4 +δ^4 +α^4)+γ^2(δ^4 +α^4 +β^4)+δ^2(α^4 +β^4 +γ^4)$

How to do the part (iv) . Please help. Here are my answers to the first parts: (i) α a root of given equation $\implies \alpha^4-5 \alpha^2 + 2 \alpha -1 = 0$ $\implies \alpha^{n+4} - 5 ...
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0answers
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Notes on theory of partial fraction decomposition

I tried searching a lot but mostly I am seeing techniques on how to decompose polynomial denominators. What I am looking for is the theory that helps me get a total picture. For example, on this link ...
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Find out the primitive polynomial GF(3)

1.) $x^2 + 2x$ 2.) $x^2 + 1$ 3.) $x^2 + 2$ 4.) $x^2 + 2x$ 5.) $x^2 + 2x + 1$ 6.) $x^2 + 2x + 2$ 7.) $x^2 $ 8.) $x^2 + x + 2$ 9.) $x^2 + x + 1$ Can any one help me in listing out primitive polynomials ...
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1answer
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Set of a summation

Let $S = \{n ∈ N | n \text{ divides the sum of any n consecutive numbers} \}$. How can I describe the set S? I was given the hint: $\displaystyle\sum\limits_{n=1}^N n=\frac{N(N+1)}{2}$ I don't want ...
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2answers
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Solving a polynomial equation by factoring

The polynomial f(x) is defined by $$f(x) = 12x^3+25x^2 -4x -12$$ (i) Show that f(-2) = 0 and factorise f(x) completely. Which i did and got $(x+2)(3x-2)(4x+3)$ (ii) Given that $$12 * 27^y + 25 * ...
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1answer
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Greatest Common Divisor of two binary polynomials

How can I find the GCD of $x^4 + x^3 + x^2 + 1$ and $x^6 + x^5 + x^4 + x^3 + x^2 + 1$? I know that $x^4 + x^3 + x^2 + 1$ is an irreducible polynomial of degree $4$, and that it is not primitive, but ...
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2answers
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Inductive proof of the degree of a polynomial

Here is the problem: Assume that there is a polynomial $P(x)$ of degree 4 such that for all $N \in \mathbb{N}$, $$P(N) = \sum\limits_{n=0}^N n^3$$ Find the polynomial. Use induction to prove that ...
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1answer
36 views

Proving that polynomials with rational coefficients have integer roots

Obviously, polynomials with integer coefficients will satisfy P(x)$\in$ Z or every x $\in$ Z. But how do we prove that those with rational coefficients can produce integer roots? For instance, I have ...
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1answer
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Factoring binary polynomials

I need to factor two binary polynomials and present each as a product of powers of irreducible polynomials. a) x⁴ + 1 I have figured it out this far: x⁴ = (x²)² and 1 = 1² So I have something in ...
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1answer
13 views

Polynomial rings of two variables

Prove that $(x,y)$ is not a principal ideal in $\mathbb{Q}[x,y]$. Here what is the definition of $(x,y)$? I don't know how to start the solution since I don't know the meaning of $(x,y)$.
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0answers
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Looking for proof of formula in WolframMathWorld article [duplicate]

I came across the formula (24) in the WolframMathWorld article on Web page http://mathworld.wolfram.com/TrigonometryAngles.html where no source of the proof could be identified by me. The formula is ...
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1answer
23 views

Remainder theorem thinking question given properties of the original equation

Consider a cubic polynomial function $y=f(x)$ with the following properties: $f(x) \ge 0$ only for $x=-1$ and $x\ge3$ when $f(x)$ is divided by $(x-4)$ the remainder is $50$. Find the equation ...
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1answer
19 views

quartic polynomial with no x-intercepts

What is an example of a 4th degree polynomial with no x-intercepts. I have looked everywhere but can not find one.
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2answers
22 views

How to solve higher grade polynomials of complex numbers $q^{10}-2q^5+2=0$

If I wanted to find the roots for $q^{10}-2q^5+2=0$, how would I go about doing that? I tried treating it like a quadratic equation, but couldn't get there. I also tried putting $q=(a+ib)$ but that ...
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2answers
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Solving a Complex Number polynomial problem

This is an example Complex equations problem, everything is well understood except --(ii) in the below solution. Please can anyone explain, how anyone could have guessed the expansion in (ii) of the ...
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2answers
14 views

Quick way to determine existence of integral root of a polynomial in one variable

Suppose $p(x) \in \mathbb{Z}[x]$ and if there exist $b \in \mathbb{Z}$ s.t. $p(b)=0$, then $x-b|p(x)$. The other technique can be to put all $b \in \mathbb{Z}$. But this require to check every $b \in ...
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2answers
23 views

Factoring Polynomial with Complex Coefficients - Cauchy's Theorem

I'm faced with another polynomial (with complex coefficients) that I seem to only know how to solve using wolfram alpha. Here is the original integral that I need to compute using algebra and Cauchy's ...
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0answers
8 views

Multiplication of polynomials in Chebyshev basis

For polynomials in the monomial basis like $p_n(x) = \sum_{k=0}^N a_k x^k $, the product of 2 polynomials is can be either found though the convolution of the 2 corresponding polynomial vectors or ...
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2answers
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If a polynomial is zero on a field F, is it zero on every extension of F?

Let $p$ be a univariate polynomial over a field $F$, and let $K$ be an extension of $F$. If $p(x) = 0$ for all $x \in F$, does this imply that $p(x) = 0$ for all $x \in K$? How about if $p$ is ...
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0answers
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Solving the characteristic equation $a^4+2a^3+5a+8=0$

I need to find the eigenvalues of a $4\times4$ matrix. I already determined the characteristic equation, which is $a^4+2a^3+5a+8$. Now I have to solve $a^4+2a^3+5a+8=0$, but I don't know how to ...
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1answer
62 views

Using induction for $x^n - 1$ is divisible by $x - 1$

Prove using induction that for all non-negative integers n and for all integers $ x > 1 $, $ x^n - 1 $ is divisible by $ x - 1 $. Step 1: We will prove this using induction on n. Step 2: Assume ...
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how to calculate the RicciTensor for system of 3 polynomials equations in 3 variables to test whether is spherical [on hold]

how to calculate the RicciTensor for system of 3 polynomials equations in 3 variables i find examples using differential expression, how to do for system of polynomial equations Riemannian metric of ...
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3answers
48 views

How many multiplications at a minimum must be performed in order to calculate this polynomial

How many multiplications at a minium must be performed in order to calculate the polynomial expression : $$x^{4} - 2x^3 + 5x^2 + x - 6 $$ Does this question mean I have to shorten the expression ...
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1answer
19 views

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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0answers
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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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0answers
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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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1answer
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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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1answer
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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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1answer
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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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1answer
28 views

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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1answer
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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
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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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2answers
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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.
6
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2answers
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
51 views

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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2answers
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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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2answers
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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 ...