This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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1answer
58 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 ...
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1answer
29 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
30 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 ...
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5answers
50 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
50 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
138 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
34 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
35 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 ...
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2answers
26 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
60 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 ...
1
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1answer
73 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
35 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 ...
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2answers
70 views

Using descartes rule of sign

Use Descartes' rules of signs to discuss the possibilities for the roots of each equation. Do not solve equation. $$p(x)= x^3+5x^2+7x+1=0$$ $p(x)$ I saw no sign change $p(-x)$ I saw 2 sign ...
2
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1answer
35 views

Expression for polynomial in $k[x,y]$.

Let $k$ be any field. For any polynomial $f \in k[x,y]$ apparently one can write $f(x,y) = f_0(x) + f_1(x)y + (x^3 - y^2)g(x,y)$. Why is this the case?
0
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1answer
165 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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3answers
110 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
62 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
118 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 ...
7
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1answer
107 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, equation (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
18 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 ...
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1answer
215 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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1answer
211 views

Ladder against a wall. [duplicate]

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
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1answer
31 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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2answers
181 views

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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2answers
528 views

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 ...
1
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1answer
16 views

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 ...
1
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1answer
30 views

Solving Yoshida equations

I want to solve $a$, $b$ and $c$ out of the following set of equations \begin{cases} a + b + c = 1 \\ a^{p+1} + b^{p+1} + c^{p+1} = 0 \\ a = c \\ \end{cases} where $p$ is even. But I absolutely have ...
0
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1answer
21 views

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 ...
1
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1answer
44 views

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 ...
6
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2answers
151 views

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 ...
0
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5answers
58 views

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
52 views

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 ...
4
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1answer
107 views

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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2answers
145 views

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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0answers
174 views

Is solving the quintic the obstacle to solving the Riemann hypothesis?

Mathematica knows how to solve: ...
1
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1answer
80 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
93 views

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
255 views

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 ...
1
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4answers
546 views

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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1answer
118 views

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 ...
0
votes
1answer
37 views

Primitive polynomial and divisibility

Let $f(x) \in \mathbb Z[x]$ with $c(f)=1$ and $f$ is non-constant. Now suppose $h(x) \in\mathbb Z[x]$ be such that $h(x)=f(x)q(x)$ where $q(x) \in\mathbb Q[x]$. Then I have to show that $q(x) ...
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1answer
149 views

In what base does the equation $x^2 - 11x + 22 = 0$ have solutions $6$ and $3$?

If we have below equation and know that $6$ and $3$ are answers of this equation, how to obtain the base used in the equation? $$x^2 - 11x + 22 = 0$$ Partial result The base is not $10$. (Because ...
0
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1answer
62 views

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 $p$ ...
5
votes
2answers
192 views

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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2answers
70 views

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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2answers
63 views

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 ...
2
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1answer
118 views

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 ...
2
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3answers
100 views

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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1answer
69 views

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 ...
0
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2answers
786 views

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$ $\operatorname{lcm}(f(x),g(x))=x^5+x^3+x^2+1$ ...