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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4
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2answers
24 views

Dividing a polynomial with $(x^2+1)^2$

I have been given that a polynomial $f(x)$ with real coefficients is divisible by $(x^2+1)$, and that when $f'(x)$ is divided by $(x^2+1)$, we get a remainder of $(x+1)$. I need to prove that ...
0
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0answers
35 views

Solving equations, Math olympiad, using vieta relation?

So the question asks to solve for real valued $a$ such that $b,c,d\in\mathbb{R}$ $$abcd=-1$$ $$(a+b)(c+d)=-1$$ $$ac+bd+a+b+c+d=-1$$ $$ab+cd=ac+a+c$$ So assuming the four numbers are roots of a quartic ...
7
votes
1answer
41 views

How do I find a constant for a polynomial so its roots are reflective around a linear function?

How can I find all complex numbers $w$ so that the roots of the following polynomial are reflected around a linear function $f(x)$ $$p(q) = q^2-4q+w = 0$$ If I want to find all the complex numbers ...
1
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0answers
17 views

Prove polynomial f(x) irreducible over the integers.

Consider the polynomial $f(x) = (x − 1)(x − 2)···(x − (n - 1))(x − n) + 1$, for some $n \in \mathbb{N}$. Prove that $f(x)$ cannot be reduced to the form $f(x) = a(x) \cdot b(x)$ for polynomials $a, b$ ...
1
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1answer
18 views

Third degree polynomial with unknown coefficients $q^3-3aq^2+b^2q+c = 0$

For an equation $q^3-3aq^2+b^2q+c = 0$ we know the roots $c, (a+b), (a-b)$. What is a good place to start with such equations? I've tried setting up a system of equations, but this is supposed to be ...
0
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1answer
19 views

Finding maximum of the basic Bernstein Polynomials

The basic Bernstein Polynomials $B_{n, k}$ are defined for all integers $n, k$ with $0 \leq k \leq n$ by $B_{n, k} = {n \choose k} x^k (1 - x)^{n-k}$ for $x \in [0,1]$. I want to prove that the ...
1
vote
1answer
44 views

$f(x)$ is still irreducible

Let $f(x) \in K[x]$ an irreducible polynomial of $K[x]$ of degree $n$. Let $K\leq F$ a field extension with $[F:K]=m$. If $(n,m)=1$ show that $f(x)$ stays irreducible also as a polynomial of ...
0
votes
0answers
18 views

Binary Polynomial Factoring

I just need confirmation that I've done my math right. If $a(x) = x^4 + x^3 + x + 1$ and $b(x) = x^2 + x + 1$ are binary polynomials, find binary polynomials s(x) and r(x) such that $x^4 + x^3 + x + ...
0
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0answers
23 views

Silly technical question about polynomials in Lagrange's “résolution algébrique”

I decided that I'd go through Lagrange's "Sur la Résolution Algébrique des Équations" (see, if you have the time, a previous, unanswered question of mine: Value in retracing mathematicians' steps ...
0
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0answers
14 views

How can we show that $T(x)= q(x)Q(x) +r(x)*R(x)$?

Concerning that Polynomial $T(x)$ is the GCD of polynomials $Q(x)$ and $R(x)$, show that there are polynomials $q(x)$ and $r(x)$ such that $$T(x)=q(x)Q(x)+r(x)R(x)$$ What I did is that according to ...
0
votes
1answer
12 views

Primitive elements of GF(8)

I'm trying to find the primitive elements of GF(8), the minimal polynomials of all elements of GF(8) and their roots, and calculate the powers of α^i for x^3 + x + 1. If i did my math correct, I ...
2
votes
4answers
65 views

Solving $y^2 - yx - y + x = 0$ for $y$?

I solved this equation for $y$ by inspection and confirmed it with Wolfram Alpha - $y^2 - yx - y + x = 0$ I got the values $y = 1$ and $y = x$ However I was wondering is there a formal method for ...
1
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1answer
35 views

Compute the degree of the splitting field

I need to compute the degree of the splitting field of the polynomial $X^{4}+X^{3}+X^{2}+X+1$ over the field $\mathbb{F}_{3}$. Quite honestly I don't really know where to begin, I know the polynomial ...
3
votes
1answer
28 views

Exact value of polynomial at trigonometric argument

Given that $$\cos 8\theta= 128\cos^8 \theta −256\cos^6 \theta +160 \cos^4 \theta −32\cos^2 \theta +1$$ Find the exact value of: $$4x^4 −8x^3 +5x^2 −x$$ where $x=\cos^2 ...
1
vote
1answer
50 views

Is there a quicker way to write $\cos (n\theta)$ in terms of $\cos \theta$?

Im writing $\cos 8\theta$ in terms of $\cos \theta$ using De Moivre's Theorem $$\cos 8\theta= \Re {(\cos\theta+ i \sin \theta)^8}$$ Let $s=\sin \theta$ and $c=\cos \theta$ $$=c^8 ...
0
votes
4answers
52 views

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.
2
votes
1answer
19 views

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 ...
0
votes
0answers
24 views

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 ...
0
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1answer
18 views

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 ...
-1
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1answer
23 views

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 ...
0
votes
2answers
22 views

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 * ...
0
votes
1answer
28 views

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 ...
1
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2answers
30 views

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 ...
1
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1answer
41 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 ...
0
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1answer
13 views

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 ...
0
votes
1answer
18 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
26 views

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 ...
4
votes
5answers
42 views

Tool for expressing $x=f^{-1}(y)$ if $y=f(x)$ is given

I have many equations of nature - $y=ax^{12}+bx^5+5x^4+1$ For all these equations, I need to express x in terms of y. What tool or software would you recommend for this? I would much prefer to ...
1
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1answer
24 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 ...
0
votes
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.
1
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2answers
24 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 ...
0
votes
2answers
21 views

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 ...
0
votes
2answers
15 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 ...
0
votes
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 ...
0
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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 ...
3
votes
2answers
31 views

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 ...
0
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0answers
38 views

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 ...
0
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1answer
63 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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0answers
13 views

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 ...
0
votes
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 ...
1
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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 ...
0
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0answers
11 views

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 ...
0
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0answers
16 views

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 ...
6
votes
1answer
77 views

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 ...
0
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1answer
11 views

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 ...
0
votes
1answer
13 views

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?
0
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1answer
32 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.
0
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0answers
15 views

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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votes
1answer
51 views

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 ...
1
vote
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 + ...