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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1
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1answer
31 views

Polynom as sum/product of symmetric polynoms

I have a polynom $(x_1^2x_3 + x_2^2x_1 + x_3^2x_2)(x_1^2x_2 + x_2^2x_3 + x_3^2x_1)$ and I need to express as sum/product of elemental symmetric polynoms $s_1,s_2,s_3$. I know there is an algoritm for ...
1
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2answers
143 views

Using Chebisev polynomials to express sin(nx) & cos(nx) as polinomials of sin(x) and cos(x)

$Sin(nx)$ and $cos(nx)$ can be expressed as polynomials of sin(x) and cos(x). I am interested in the way of this expression and a proof (preferably at secondary-school level) as well.
2
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0answers
358 views

Eigenvalues of 5x5 matrix given equation involving matrix

I have been given the matrix $A$ and we are told it is a $5\times 5$ matrix s.t. $A^4=A^2\neq A$. I want to find the eigenvalues so I tried $A^2(A-I)(A+I)=0$ so the eigenvalues are $0, 1, -1$ but I ...
1
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0answers
47 views

To prove given $ r \cdot f_1+f_2\cdot (s+1)$ one who knows $f_2$ cannot find out what $f_1$ is

We define the polynomials $r,f_1,f_2,s\in R[x]$. Where $r$ is a random degree 1 polynomial and $s$ is a random polynomial such that: $\deg(s)=\deg(f_1)=\deg(f_2)$. Let $R$ be $\mathbb {Z}_q$ where $q$ ...
1
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1answer
175 views

How to solve: $x^4+x^2=1$

I solved $x^4+x^2+1=0$. But, the above one is hard. The equation is too hard for me to understand. Can anyone solve it? Please help.
0
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1answer
45 views

An $\Bbb{R}\to\Bbb{R}$ function with two plateaus of different heights and a valley

I am looking for a $\Bbb{R}\to\Bbb{R}$ function $f$ with two plateaus of different heights and a valley. The function has a minimum for $x=a$ and $f(a)=b$. The first (the one for smaller $x$) ...
1
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1answer
44 views

Polynominal odd function [closed]

If $f(x$) is an odd function and $x-y$ is a factor. show that $x^2-y^2$ is a factor as well I'm having trouble to solve this
1
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1answer
50 views

Writing a particular polynomial as product of irreducibles in various rings.

I want to factor the polynomial $x^3-10x+4$ into a product of irreducibles over each of the fields $\mathbb{Z}[i]$,$\mathbb{Q}[\sqrt{2}]$, $\mathbb{Q}[\sqrt{2},\sqrt[3]{2}]$, $\mathbb{Z}/ 11 ...
1
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0answers
64 views

Prove that $p(x)=(x-1)(x-2) \cdots (x-n) + 1$ is irreducible over $\mathbb{Z}$ for all $n \geq 1$, $n \neq 4$. [duplicate]

Prove that $p(x)=(x-1)(x-2) \cdots (x-n) + 1$ is irreducible over $\mathbb{Z}$ for all $n \geq 1$, $n \neq 4$. I do not clearly see how to solve this problem and what is so special about the integer ...
2
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4answers
132 views

Solve the following equation: $\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$

Solve the following equation: $$\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$$ I know it's from a Math Olympiad but I don't know which and I couldn't find it on the internet. Expanding everything ...
1
vote
2answers
204 views

$F$ field, $\alpha$ separable on $F$. Is $F(\alpha)$ a separable extension of $F$?

Let $F$ be a field, and let $\alpha$ be algebraic and separable over $F$. Is $F(\alpha)$ a separable extension of $F$? By "$\alpha$ is separable" I mean that its minimum polynomial over $F$ is ...
4
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0answers
50 views

How to find out if a polynomial equation has real solutions?

I have a polynomial equation of $N$th order. The coefficients of the equation are parametrized by two variables, let's call them $a$ and $b$, both of which are real and positive. For general $N$, I ...
2
votes
1answer
16 views

Differential Equation involving Polynomial Discriminants

So this is a homework question in my algebra class that I'm getting really stuck on... it should be straightforward, but I'm not sure how to interpret the differential equation. Any hints (solutions ...
1
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1answer
96 views

Fourth degree polynomial with rational coefficients and a real root

If a quartic has rational coefficients and one real root, how would one go about showing that the real root is rational? I understand that the condition is equivalent to showing that having a ...
4
votes
1answer
73 views

How to show that the polynomilal $\sum_{k=0}^n \dfrac 1{3^{k^2}}x^k$ has $n$ distinct real roots for any positive integer $n$ ?

From this Rational roots of polynomials ; How might we show that the polynomilal $\sum_{k=0}^n \dfrac 1{3^{k^2}}x^k$ has $n$ distinct real roots $\forall n \in \mathbb N$ ?
8
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1answer
232 views

Existence of a Polynomial

Does their exist a non-linear polynomial $P(x)$ such that for every rational number $y$ there exists a rational number $x$ such that $y=P(x)$?
0
votes
1answer
111 views

Is Rufinni's rule the quickest hand-method to find roots in high order polynomials?

I'm wondering if there's another method where I do not have to "trial-error" with every guess neither using numerical methods. When the searched root is 3*π/5 with Ruffini's rule I cannot find it ...
0
votes
1answer
61 views

gcd of polynomials over Z_7

I want the gcd of the two polynomials: $$f=x^5+3x^4+5x^3+x^2+x+3$$ $$g=2x^3+4x^2+x$$ in $Z_7[x]$. My approach: I use the euclidean algorithm and continue until I get no remainder. ...
0
votes
1answer
164 views

Find exact value of $\cos (\frac{2\pi}{5})$ using complex numbers.

Factorise $z^5-1$ over the real field. Show that $\cos \frac{2\pi}{5}$ is a root of the equation $4x^2+2x-1=0$ and hence find its exact value. I have worked out that $$ ...
12
votes
6answers
294 views

If the number $x$ is algebraic, then $x^2$ is also algebraic

Prove that if the number $x$ is algebraic, then $x^2$ is also algebraic. I understand that an algebraic number can be written as a polynomial that is equal to $0$. However, I'm baffled when showing ...
33
votes
2answers
815 views

Are most rational quintics unsolvable?

It is well-known that, as polynomials of degree exceeding 4, there exist quintics whose roots cannot be solved for by radicals (Abel-Ruffini theorem). So we can divide the set of rational quintics ...
6
votes
2answers
299 views

Can we make a sequence of real numbers such that polynomial of any degree with co-efficients of the sequence has all its roots real and distinct ?

Does there exist a sequence of real numbers $(a_n)$ such that $\forall n \in \mathbb N$ , the polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_o$ has all $n$ real roots ? Can we make a sequence so that all ...
1
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1answer
47 views

GCD of polynomials by using Euclid's algorithm

Let $g = x^2 +6x -7$ and $f = x^4 - 1$. Find the GCD of $f$ and $g$. So I started by evaluating $f/g$ and the result is $q = x^2-6x+43, r = -300x+300$. I tried to follow the algorithm one step ...
1
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0answers
21 views

Localization of polynomial ring as differentiable functions

Let $a \in \mathbb{R}$ be a point and $S=\mathbb{R}[x]_{(x-a)}$ the localization of the polynomial ring $\mathbb{R}[x]$ with maximal ideal $(x-a)$. i) Describe the elements of $S$ as differentiable ...
0
votes
1answer
10 views

The number of polynomial in a polynomial ring

If we define a poylnomail ring $R[x]$, I need to know the number of polynomial (in this ring) of a particular degree, $d$, please.Let $R=Z_P$, where $p$ is a prime number.
3
votes
4answers
165 views

How do I find the sum of the cubes of the roots in a cubic polynomial?

I have an equation, $x^3-x^2+x-2$, with three distinct roots, $p$, $q$ and $r$. What is the value of $p^3+q^3+r^3$? I'm not sure how to do this. Using Vieta's formula, we know that: $pqr= 2$ ...
0
votes
2answers
85 views

Second Degree Polynomial Interpolation, error related

We want to create a table of the exponential integral function $$E_{1}(x)=\int_{x}^{\infty}\frac{e^{-t}}{t}dt, x>0$$ over the interval $x \in [1,10]$ with stepsize $h$. How large can $h$ be if a ...
1
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2answers
38 views

It is div:$\mathbb{P}_k\rightarrow \mathbb{P}_{k-1}$ surjective?

My question is: It is the operator $\text{div}:\;(\mathbb{P}_k){\color{red}{^3}}\rightarrow (\mathbb{P}_{k-1})$ surjective ? Here $\mathbb{P}_k$ denotes, as usual, the set of polynomial with degree ...
2
votes
0answers
63 views

Does the equation $\tan(x)=y$ have any non-zero rational solution?

Trivially $\tan(0)=0$ but it seems this is the "unique" solution of the equation $\tan(x)=y$ on rational numbers. In fact if we try to make $y$ rational we usually use irrational (transcendental) ...
0
votes
1answer
38 views

synthetic division/long division divisor sign

I know that if you are dividing by $x-3$ with long, then if you do it with synthetic division it's going to be positive 3 that gets used, the value used in synth. division is opposite. So if your ...
1
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0answers
45 views

Solvability by radicals of Polynomials defined by a recurrence relation

I want to determine the smallest integer $m$ such that the polynomial $P_{n}(x)$, $n\geq m$, given by : $$\left \lbrace \begin{array}{l} P_{n+1}(x) = P_n(x) (x-n-1) + \prod\limits_{i = 0}^n x-i\\ ...
2
votes
3answers
87 views

Finding roots of a quartic

How do I find the roots of the equation $$(x+3)^5-(x+1)^5=7$$ I tried opening it up, it turns into a ugly quartic which doesn't factor. I don't know what to do next. Please help me out.
3
votes
2answers
57 views

Show that no linear polynomial divides $x^k + x^{k-1} + \cdots + 1$ with $k\ge 2$ even

Let $f(x) = x^k + x^{k-1} + \cdots+ 1 \in \mathbb{Q}[x]$, $k\ge 2$ and even. Show there's no linear polynomial which divides $f(x)$. A start: Lets assume by contradiction there's a linear ...
3
votes
5answers
87 views

Factorization of $x^6 - 1$

I started by intuition since I'm familiar with the formula $a^2-b^2 = (a-b)(a+b)$. So in our case $$x^6 - 1 = ({x^3} - 1)({x^3}+1)$$ How should I proceed? I assume there's some sort of algorithm to ...
5
votes
1answer
129 views

Solve $x^5 + x - 1 = 0$

Solve $x^5 +x - 1 = 0$ I am simply curios to see how the solution would go, since it is a quintic, it cannot be done by regular methods. Im just curios to see what people come up with (I can't solve ...
1
vote
1answer
56 views

Eigenvalues, polynomials and minimal polynomials

I have proved (a) by: Let $\lambda$ be an eigenvalue of $AB$ $ABv=\lambda*v$ Then $BABv=\lambda*B*v$ so Bv is an eigenvector of BA with eigenvalue $\lambda$. For B, I have found the formula in ...
4
votes
3answers
152 views

The inequality $x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+3/4 >0$ holds for all $x\in\mathbb R$

Show $\forall \ x \in \mathbb{R}:\quad x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+\dfrac{3}{4}>0$ My attemps: Case $x=-1$ That is true for this case Then for $x \neq - 1$: $$\dfrac 3 4 - x + ...
1
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0answers
183 views

Does synthetic division and long division of polynomials always give the same remainder?

Does synthetic division and long division of polynomials always give the same remainder? On a couple of the problems I did, I did, but now I've got a couple that are coming out different and I don't ...
0
votes
1answer
25 views

Long division, polynomials

So I've been find doing ones where the polynomial exponents all counted down without missing a number, like: $x^3 + x^2 + 8x$, bu what do you do when it skips one, like: $x^4 - x^2 + x$
3
votes
3answers
148 views

Finding the zeroes in a function

I have come across a problem on my trigonometry homework where we need to find the zeros of a function without the use of a calculator. The Equation: Given one of the zeros is $x=5$ $f(x) = ...
1
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1answer
51 views

Differentiation map and the Cayley-Hamilton theorem

I have computed (a) to be $-\lambda^3$. I also know that the Cayley-Hamilton theorem states that substituting the matrix A (where A is matrix with p(λ)=det(λI-A) for λ in this polynomial results in ...
2
votes
4answers
95 views

can have solution of $x^4-3x^3+2x^2-3x+1=0$ using only high school methods

can have solution of $x^4-3x^3+2x^2-3x+1=0$ using only high school methods??? i only know quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ i tried many algebraic manipulations and i get ...
0
votes
3answers
81 views

When we divide P(x) with (x-1) the remainder is 2, when we divide it with (x-2) it is 5. We divide it with (x-1)(x-2) the remainder is ax+b.

Find a and b. ACCORDING TO ME since the remainder is ax+b it seems that the polynomial has no x^1. I've tried to express it with using a, b, c, d, e but there were many variable to solve it. Please ...
6
votes
4answers
98 views

Find all polynomials with real coefficients that satisfy $(x^2-6x+8)P(x)=(x^2+2x)P(x-2)$

Find all polynomials with real coefficients that satisfy $$(x^2-6x+8)P(x)=(x^2+2x)P(x-2)\forall x\in\Bbb R$$ My work; $$\frac{P(x)}{P(x-2)}=-\frac{4}{x-2}+\frac{12}{x-4}+1\tag{1}$$ ...
1
vote
2answers
59 views

Question on reducibility over rationals.

If we have a polynomial with real coefficients $P(x)$ for which $P(x)=q$ has a rational solution for all rational numbers q, does that mean that the coefficients actually have to be rational? I think ...
1
vote
1answer
21 views

If $f \mid h, g\mid h$ and $f,g$ are relatively prime, then $fg\mid h$?

Let $f,g,h \in\mathbb{F}[x]$, with $f$ and $g$ are relatively prime. If $f\mid h$ and $g\mid h$, prove that $fg\mid h$. What I've done so far: Experimenting with natural numbers, I suspect that ...
3
votes
3answers
400 views

Eigenvalues of operator $p(T)$ in terms of the eigenvalues of $T$, where $p$ is a polynomial

Let $T$ be a linear operator on a finite dimensional vector space over an algebraically closed field $F.$ Let $f$ be a polynomial over $F.$ Prove that $c$ is a characteristic value of $f(T)$ iff ...
6
votes
0answers
67 views

Does each irreducible polynomial over the integers represent at least one prime? [duplicate]

There's not much more to the question: If $f(x) \in \mathbb{Z}[x]$ is an irreducible polynomial, is there a simple proof that there must be some $x$ for which $f(x)$ is a prime number (positive or ...
2
votes
5answers
795 views

How do you find the imaginary roots of a fourth degree polynomial that cannot be simplified?

I started out with $f(x)=16x^6-1$, and I got it down to $64x^4+16x^2+4$ by synthetically dividing by roots $0.5$ and $-0.5$ How should I continue in order to find the other roots?
1
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0answers
60 views

Expression for a polynomial

Let $s\in\Bbb R[x_1,\dots,x_n]/(x_1^2-x_1,x_2^2-x_2,\dots,x_n^2-x_n)$ and supposing for every $A\subseteq \{1,2,\dots,n\}=[n]$ we fix $\alpha_A\in\Bbb R$ and $$s\equiv\alpha_A\mod (x_u,x_v-1)_{\forall ...