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
19 views

Relation between polynomials.

Let $P_{\lambda}(X) = X^4 + 6X^2 + \lambda X -3$ be a polynomial for every $\lambda \in\mathbb{C}$. Prove that if $\alpha \in \mathbb{C}$ is a root of multiplicity $2$ of $P$, then it is a root of ...
0
votes
1answer
59 views

Injective polynomial on the unit disc

Let $P(z)=\sum_{k=0}^{n}{a_kz^k}$ be polynomial that is injective in the open unit disc. Show that $|a_n|\le |a_1|/n$. I know that if $P$ is injective function than $P$ is conformal map and therefore ...
0
votes
2answers
129 views

Solving polynomial equation using known trigonometric identity

I recently did an exam paper in which the following question was asked: Prove $$\sin(5\theta)=16\sin^{5}(\theta)-20\sin^{3}(\theta)+5\sin(\theta)$$ Hence find all the solutions to: ...
1
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1answer
47 views

Expanding brackets

My question today is whether or not it is in mathematical convention to do the following when expanding brackets. If we have the expression $(x+2)(x-5)$ and we expand this out, we get $x^2-3x-10$. ...
1
vote
0answers
69 views

Q th order polynomial transform to represent all the curves in $\mathbb{R^d} $

In space $ \mathcal{X} = \mathbb{R^2} $, to get all possible quadratic curves in $ \mathcal{X} $, we need feature transform $\mathbf{z} = \Phi_2(\mathbf{x})$, where $\mathbf{x} \in \mathbb{R^2}$, and ...
2
votes
5answers
86 views

Prove that $p \in \mathbb{R}[x]$ can be represented as a sum of squares of polinomials from $\mathbb{R}[x]$

$p \in \mathbb{R}[x]$ and $ \forall x\in\mathbb{R} \ \ p(x) > 0$. Prove that $p = \sum_{k=1}^{n}p^2_k, \ \ p_k \in \mathbb{R}[x]$. I have noticed that $deg(p)$ must be even, because $p(x)$ doesn't ...
12
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1answer
295 views

Galois group of the quintic polynomial $X^5+X+1$

I'm trying to find the Galois group of the polynomial $p(X)= X^5+X+1$ over $\mathbb Q$. First, one notes that, if $\omega$ is a primitive cubic root of unity, then it is a root of $p(X)$. So, ...
0
votes
7answers
435 views

How to derive $\;a^3 + b^3 = (a + b)(a^2 - ab + b^2)\;?$

I'm studying in 8th grade in India. I saw the question in my FIITJEE textbook, but it was not shown how to derive it. Please help. My mid phase examinations are coming up.
3
votes
1answer
169 views

Angle between two polynomials

Given the inner product of two polynomials $p(X), q(X) \in P(d)$, where $P(d)$ is the vector space of all polynomials of degree less than or equal to d, with real coefficients, and using the inner ...
0
votes
1answer
195 views

Writing a piecewise polynomial function as a sum of truncated power functions

Writing a piecewise polynomial function f(t) as a sum of truncated power functions p(t)= (t-c)^k where f(t) is defined as $$ f(t) = \begin{cases} 0 &, 0 \leq t < 1 \\ t - 1 ...
1
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2answers
65 views

A polynomial formula for the primes

Is there a proof that there is no polynomial which would return $n$th prime for the input value $n$? In other words is there an explanation for why there is no polynomial $P(x)$ such that $P(n)=p_n$ ...
11
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0answers
205 views

When does a polynomial fixing a subring imply its coefficients are in that subring?

Let $S$ be a subring of $R$. If $p$ is a polynomial with coefficients in $S$, then $p$ fixes $S$ (as a function, that is, $p(s)\in S$ for all $s\in S$). A converse statement is: If $p$ is a ...
6
votes
3answers
90 views

Solving Equation of Degree n, where n is any value between 1 and 2

How does one solve an equation of the form: $$ax^n + bx + c = 0$$ where n is a non integer value between 1 and 2. Is there a formula to provide an analytic solution?
0
votes
1answer
27 views

Solve for simple monotonically increasing polynomial

What would be the most programmatically efficient way of solving an equation that looks like: x + x^2 + x^3 + ... + x^N = y There is never any coefficients, and Y will always positive and known. I ...
3
votes
1answer
34 views

Existence of a certain polynomial in $\mathbb Z [X,Y]$

I am at a point where I need to know whether there is a polynomial $f \in \mathbb Z [X,Y]$ such that: $f(1,y) \ge 0$ for all $y \ge 0$ $y-1,x \ge 0 \wedge f(x,y) \ge 0 \Rightarrow 0 \le f(2x,y-1) ...
0
votes
2answers
92 views

Are there infinitely many polynomials crossing through a finite amount of points?

If I have $3$ $(x,y)$ points, say $(2,3)$, $(8,17)$ and $(20,25)$, how many polynomials are there that pass through those $3$ points? Infinitely many or a finite amount? What if I have $n$ arbitrary ...
4
votes
4answers
455 views

Prove that p has m distinct roots if and only if p and p' have no roots in common

Problem: Suppose $p \in \mathcal{P}(\mathbf{C})$ has degree $m$. Prove that $p$ has $m$ distinct roots if and only if $p$ and its derivative $p'$ have no roots in common. My proof so far: If $m=0$, ...
5
votes
2answers
46 views

The relationship between the intercepts and the remainder in the remainder theorem

The polynomial remainder theorem states that when a polynomial $P(x)$ of degree $> 0$ is divided by $x-r$ ($r$ being some constant) the remainder is equal to $P(r)$, that is: $$\begin{array}l If ...
4
votes
1answer
106 views

Proving $\cos x < 1 - \frac{x^2}{2} +\frac{x^4}{24}$

I wish to prove the following inequality for $x\ne 0$: $$\cos x < 1 - \frac{x^2}{2} +\frac{x^4}{24}$$ Using the fact that I already prove: $$\cos x > 1 - \frac{x^2}{2}$$ My try: $\cos x = 1 - ...
33
votes
4answers
1k views

$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x)$, then all the roots of $p_k(x)$ are real

$p_0(x)=a_mx^m+a_{m-1}x^{m-1}+\dotsb+a_1x+a_0(a_m,\dotsc,a_1,a_0\in\Bbb R)$ is a polynomial, and $$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x),\qquad n=1,2,\dotsc$$ then, there exist $N\in\Bbb N$, such ...
0
votes
1answer
57 views

Prove that if $p$ is prime then $\frac{X^p-1}{X-1}=X^{p-1}+\cdots+X+1$ is irreducbile in $\mathbb{Q}[X]$. [duplicate]

I have no idea how to approach this. I'm supposed to use $f\in \mathbb{Q}[X]$ is irreducible $\iff \exists{a}\in\mathbb{Q}$ such that $f(X+a)$ is irreducible. I tried to use $a = 1 \in \mathbb{Q}$ ...
2
votes
1answer
249 views

Primitive polynomials in LFSRs

I need help proving the following theorem. I found it many books but on every single one it says that they omit the proof because it is in every good textbook. THM Let $c(x)$ be a connection ...
1
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2answers
113 views

A set with zero density

Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that $$n \mid a^{f(n)}-1.$$ Prove that $S$ has density $0$; that ...
0
votes
1answer
48 views

Polynomial equal to analytic function

Let p(z) be a polynomial of degree n. Show $\exists R$ and analytic $f(z)$ such that $p(z)=f(z)^n$ for $|z|>R$
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votes
2answers
36 views

Show that $p+q^{2}=1$ where $x^{3}+px+q=0$ and one of the roots is the reciprocal of the other?

let the three roots be $z, 1/z, t$. So $z+1/z+t=0$ and $zt+1+t/z=p$ and $z(1/z)t=-q=t$ $-1/z-z=t$ $p+q^{2}=zt+1+t/z + t^{2}$ How do I simplify the RHS to get 1?
3
votes
0answers
52 views

Prove that the set has zero density

Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that $$n \mid a^{f(n)}-1.$$ Prove that $S$ has density $0$; that ...
1
vote
1answer
80 views

Indefinite integration of general polynomial. Is this correct?

I was reading some notes of a guy I was tutoring the other day on basic calculus. He noted that if $$\int{x^n dx}=\frac{x^{n+1}}{n+1}+c,$$ then that can be extrapolated to all polynomials. He wrote, ...
5
votes
4answers
203 views

Solve $x^{3}-3x=\sqrt{x+2}$

Solve for real $x$ $$x^{3}-3x=\sqrt{x+2}$$ By inspection, $x=2$ is a root of this equation. So, I squared both sides and divided the six degree polynomial obtained by $x-2$. Then I got a ...
6
votes
2answers
1k views

Proof that no polynomial with integer coefficients can only produce primes [duplicate]

Doing a discrete math review and am trying to solve problem 1.6 in the text found here: http://courses.csail.mit.edu/6.042/fall13/ch1-to-3.pdf - I believe I've gotten parts (a) and (b) correctly, but ...
6
votes
1answer
78 views

Simple field extension and roots of a polynomial

Let $K$ be a field, $f \in K[X]$ separable and irreducible with $\text{deg}(f)=n$; $x_1,...,x_n$ are the roots of $f$ in a splitting field of $f$ over $K$. Let $g \in K[X]$ be any polynomial with ...
0
votes
1answer
62 views

Powers generate monomials

What is a reference in the literature for the following fact? Let $A$ be a commutative $\mathbb{Q}$-algebra. Then every monomial in $A$ of degree $n$ may be written as a linear combination of $n$th ...
0
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1answer
24 views

$\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$?

I want to find out if this affermation is true: let $f\in \mathbb{Q}[x]$ such that $\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$? (We know that it has ...
0
votes
1answer
60 views

Minimax polynomials?

My linear algebra text discusses the solution to minimization problems by use of the Gram-Schmidt procedure, which produces the an approximation such as $$\sin{x} = \frac{105(1485 - 153\pi^2 + ...
-6
votes
1answer
74 views

Can we abstractly construct this ring and what is it isomorphic to?

Define $R = \Bbb{Z}[X_1, X_2, \dots]$. Then place on $R$ the relations $$ X_1 + 1 = X_2, \\ X_2 + 2 = X_3, \\ X_3 + 2 = X_4, \\ X_5 + 2 = X_6, \dots \\ X_{2k-1} + 2 = X_{2k}, \ \forall k \geq 3 ...
1
vote
0answers
66 views

Different generators of (x,y) in k[x,y] give rise to automorphism.

I am stuck with the following algebra problem: Let $f,g\in k[x,y]$ be polynomials which generate $(x,y)$ (as an ideal). Consider the homomorphism $\phi:k[x,y]\to k[x,y]$ which is identity on $k$, and ...
1
vote
1answer
173 views

Finding coefficients in polynomial efficiently

Given a transitive $G$-set $ M $. I'm interested in finding the number of fixed points of $ G $ acting on $\operatorname{Pot}_a( M ) := \left\{ N\subseteq M; |N| = a \right\} $ by using the table of ...
2
votes
2answers
71 views

Nonzero nilpotent elements in $\Bbb C\otimes\Bbb Q[x]/(f)$?

I have to find if this affirmation is true: Let $f\in \mathbb{Q}[x]$ such that $\gcd(f,f')=1$, then in $\mathbb{C}\otimes_{\mathbb{Q}} \mathbb{Q}[x]/(f)$ there are no nonzero nilpotent elements. ...
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0answers
22 views

Linear Transformation - linear algebra question [duplicate]

$T:\mathbb{R}_2[x] \mapsto \mathbb{R}_2[x]$ s.t.: $$ \begin{array}{l} T(1) = 3+2x+4x^2, \\ T(x) = 2+2x^2, \\ T(x^2) = 4+2x+3x^2. \end{array} $$ Is there base $B$ of $\mathbb{R}_2[x]$ that $[T]_B = ...
0
votes
1answer
23 views

Does the root formula differ for a higher order poly?

Does the root formula differ for a higher order system? I know that a the discriminant differs for a higher order system, but does the root formula from its standard second order system form differ ...
3
votes
1answer
124 views

Polynomials vs polynomial functions

On my algebra course, sometimes we write, say $$f \in R[X], f= X^2 + X + 1$$ And sometimes we treat polynomials as functions, so $$ f(x) = x^2 + x + 1$$ What is the difference between these two ...
0
votes
1answer
89 views

Have you seen this theorem before? (GCD divides, neccessary & sufficient condition)

Conjecture. Let $a,b, c\in \Bbb{Z}, b \neq 0$, The following conditions are equivalent: (1) $d = \gcd(a,b)$ divides c. (2) There's a polynomial in $f \in \Bbb{Z}[X,Y]$ with $c$ constant term, such ...
2
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0answers
35 views

Show that $P_a(z)=0$ iff $z=N(a)$ for polynomial $P_a$

Let for $a_0=(a_0,a_1,...,a_n)\in\mathbb C^{n+1}$ the polynomial $P_{a_0}=\sum_{k=0}^na_kz^k$ and $z_0\in\mathbb C$ with $P_{a_0}(z_o)=0$ and $DP_{a_o}$ (the differential matrix) invertible. Show ...
6
votes
1answer
179 views

Factorise $x^4 + 3x^2+ 6x+ 10$

I need to factorise $x^4 + 3x^2 + 6x + 10$ completely over $\mathbb{Q}$. I am not sure how to do this. I can't find any roots of this equation in $\mathbb{Z}$.
1
vote
1answer
66 views

Exercise about basis

I am trying to solve the following exercise: Let $A \in L(P_3)$ be defined by a matrix: where $A^b_e$ = $\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$ ...
1
vote
0answers
27 views

Factorisation algorithm for polynomials in several variables over $\mathbb{Z}$.

What algorithm is used by a CAS to decide how to factor a polynomial in several variables over $\mathbb Z$?
2
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3answers
97 views

How does $\frac{t^2}{t+1}$ equal $t-1+\frac{1}{t+1}$?

I do the long division: 1: t+1 goes into $t^2$ t times 2: Subtract $t^2$ + 1 from $t^2$ and get -1 3: Answer: t - $\frac{1}{t+1}$ Am I missing something here?
5
votes
1answer
95 views

Find $\lfloor {\alpha}^6 \rfloor$

If $\alpha$ is a real root of the equation $$x^5-x^3+x-2=0$$ find the value of $\lfloor {\alpha}^6 \rfloor$. This one totally stumped me. We are asked to calculate $\lfloor {\alpha}^6 ...
3
votes
1answer
93 views

Is this definition of a polynomial adequate? If not, how do I fix it?

A function $p$ is a polynomial with coefficients in F if $p$: F $ \to$ F as $p(z) = a_0 + a_1z + a_2z^2 + ··· + a_mz^m$ for some $a_0,\ldots,a_m$ and all $z \in$ F.
0
votes
1answer
49 views

Using elementary polynomials to solve system of linear polynomials

Problem Statement I am given a finite set of monic polynomials in t, parameterized by $r_i$ $X_i = t - r_i$ where the $r_i$ are guaranteed unique. Neither $t$ nor $r_i$ are known, only $X_i$. I ...
6
votes
3answers
134 views

How to solve the following? $ x^3+1=2{(2x-1)}^{1/3} $.

Find all the real solutions of $$x^3+1=2{(2x-1)}^{1/3} $$ I tried to cube both sides but got messed up with a nine degree equation! Please help. Thanks in advance!