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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2
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1answer
178 views

When does a higher order polynomial have complex roots?

I try to say it all in the title. I'm wondering under what conditions a matrix will have complex eigenvectors and eigenvalues. That question, I think, reduces to whether the characteristic ...
3
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1answer
96 views

Prove $\sum_{i=0}^{n}\left(x_{i}^{n}\prod_{0\leq k\leq n}^{k\neq i}\frac{x-x_k}{x_i-x_k}\right)=x^n$

Suppose $x_0$ , $x_1$ , $x_2$ , ... , $x_n$ are distinct real numbers , prove that : $$\large{\displaystyle{\sum_{i=0}^{n}\left(x_{i}^{n}\prod_{0\leq k\leq n}^{k\neq ...
0
votes
1answer
205 views

Showing $f(x)$ is constant.

Let $f(x)=a_nx^n+...a_1x+a_0$ is an integer polynomial with $a_n>0,n\not=1$. $f(p)$ is prime for every $p$, where $p$ is prime. How to show $f(x)$ is constant, or not?
4
votes
1answer
150 views

Is $Z(x^2-y^3)$ isomorphic to $Z(y^2-x^3-x^2)$ over the complex numbers?

I'm having trouble determining if the algebraic sets $Z(x^2-y^3)\subset \mathbb{A}^2$ and $Z(y^2-x^3-x^2)\subset\mathbb{A}^2$ are isomorphic over $\mathbb{C}$. My guess is that this boils down to ...
0
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3answers
38 views

Last step in a proof that this set is generated by those elements

Can someone tell me, what the generator (in $\mathbb{Z}[X]$) of the ideal $T$ of all polynomials with integer coefficients, such that the first is divisible by $20$ and the second and the third are ...
0
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1answer
56 views

Show that there exists an extension of this polynomial $p(x)$ having all roots of $p(x)$

The problem: Show that $p(x)=x^2-x-1\in \mathbb Z/(3)[x]$ is irreducible over $\mathbb Z/(3)$. Show that there exists an extension K of $\mathbb Z/(3)$ with nine elements having all roots of $p(x)$. ...
1
vote
1answer
209 views

A question on the irreducible divisors and splitting field of $x^{p^n} - x\in \mathbb F_p[x]$.

I need to prove that any irreducible polynomial $f$ of degree $d\,\big|\,n$ over $\mathbb F_p$ devides $x^{p^n} - x$. I know that the splitting field of the latter is the finite field with $p^n$ ...
3
votes
2answers
663 views

Injective map on coordinate ring implies surjective?

Suppose that $f:X\rightarrow Y$ is a morphism between two affine varieties over an algebraically closed field $K$. I believe that if the corresponding morphism of $K-$algebras, ...
0
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3answers
384 views

How do I prove that a polynomial F[x] of degree n has at most n roots

It's a really basic question,in these days, I've been thinking why a polynomial $p(x)\in F[x]$ ($F$ a field) with degree $n$ can have at most n roots. It seems easy to prove, but I've been trying to ...
1
vote
1answer
236 views

Generate a polynomial w/ integer coefficients whose roots are rational values of sine/cosine?

I'm a high school calculus/precalculus teacher, so forgive me if the question is a little basic. One of my (very gifted) students recently came up with a construction yielding a quartic, one of whose ...
5
votes
0answers
165 views

Irreducibility of polynomials via Frobenius map

I am having trouble trying to show this: Let $f \in \mathbb{F}_p[x]$ be a non-constant polynomial and let $F$ denote the Frobenius map $F: R \rightarrow R$ where $R = \mathbb F_p[x]/(f)$. Prove ...
0
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1answer
46 views

how do I prove that $(1,u,\ldots,u^{n-1})$ forms a basis of $F(u)$ over $F$.

Theorem Let $p(x)$ be an irreducible polynomial in $F[x]$ and let $u$ be a root of $p(x)$ in an extension $E$ of $F$. Then if the degree of $p(x)$ is $n$, the set $(1,u,\ldots,u^{n-1})$ forms a basis ...
1
vote
1answer
214 views

irreducible polynomial of degree 2 or 3 without roots in an integral domain.

It is well-known that a degree 2 or 3 polynomial over a field is reducible if and only if it has a root. But what about integral domains? Can we have a reducible polynomial over an integral domain ...
2
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0answers
95 views

Irreducibility of polynomial

In one of my proof for my assignment I reached a point where I have to prove that $x^9-t^9$ is irreducible in $\mathbb{Z}_7(t^9)[x]$. I am unsure weather this is irreducible. If it is, how do I prove ...
5
votes
2answers
99 views

Prove that $\frac{n-1}{n}>\frac{2a_0a_2}{a_1^2}$

Given that the following equation $$p(x)=a_0x^n+a_1x^{n-1}+...+a_{n-1}x+a_n=0$$ has $n$ distinct real roots. Prove that $$\frac{n-1}{n}>\frac{2a_0a_2}{a_1^2}$$
0
votes
1answer
74 views

How do I prove that a irreducible polynomial in F[x] has a root in an extension E of F.

In order to demonstrate the root extension theorem, I need to prove that if an element of F[x]/(p(x)) is represented as $\overline a =a+(p(x))$ where $p(x)=a_0+a_1x+\cdots+a_nx^n$, then$\overline ...
2
votes
4answers
88 views

Solve for $x$ in this equation

How do I solve for $x$ algebraically? $$\dfrac{x^2(x^2-1)}{x+3} = 12$$
1
vote
2answers
211 views

Show that this polynomial is reducible

I'm trying to prove this question: Show that $p(x)=x^3 + ax^2 + bx +1 \in \mathbb Z [x]$ is reducible over $\mathbb Z$ if and only if either $a=b$ or $a+b=-2$. I did the converse in this way: if ...
2
votes
1answer
103 views

on systems of bivariate polynomial equations (quartic)

I need to find an analytical solution to a system of bivariate polynomials. Specifically: \begin{eqnarray} a_0 + a_1 x + a_2 y + a_3 xy+a_4 x^2 + a_5 y^2 + a_6 xy^2 + a_7 x^2 y + a_8 x^2 y^2 &= ...
2
votes
1answer
290 views

minimal polynomial of power of algebraic number

Consider any algebraic number $\alpha$ which is given by its minimal polynomial $f$. How can I compute the minimal polynomial of $\alpha^m$ for some natural number $m$? How efficient the algorithm is? ...
1
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2answers
174 views

About a certain type of polynomial

Let $p$ be a polynomial over the set of positive integers such that $p(n) > n$ for all positive integers $n$. It is also known that for every positive positive integer $m$ , there exists a term of ...
0
votes
2answers
2k views

Pseudo-code for Polynomial Long Division

I'm busy writing a polynomial long division class in Java, and I see that Wikipedia provides a great example for performing the long division by hand. However, when I compare it to the provided ...
4
votes
2answers
468 views

Solve logarithmic equation

I'm getting stuck trying to solve this logarithmic equation: $$ \log( \sqrt{4-x} ) - \log( \sqrt{x+3} ) = \log(x) $$ I understand that the first and second terms can be combined & the logarithms ...
2
votes
1answer
51 views

Existence of a non-variety with special properties

Does there exist an infinite set $X\subseteq\mathbb{R}^n$ such that every non-zero polynomial $P\in \mathbb{R}[x_1,x_2,...,x_n]$ has finitely many zeros in $X$?
3
votes
0answers
33 views

Study a particular polynomial sequence

Let us define the following sequence of polynomials for every two non-negative integers $i,d$: $$s_i^d(w)=\sum_{j=0}^{d+1} (-1)^j {d+1\choose j} (j+1)^i w^{d+1-j}.$$ Conjecture: The sequence ...
3
votes
3answers
1k views

Characterization of irreducible polynomials over finite fields

How much is known about irreducible polynomials over finite fields? I have seen the formula (a result of Möbius inversion) that gives the number of such polynomials, but I am looking for something ...
1
vote
3answers
48 views

Computational complexity proof

I would like to know how to prove the following: $2^n \in O(n!)$ I know that I have to show that for a constant C, we have $2^n \leq C*n!$ Right?
4
votes
3answers
874 views

Irreducibility of $X^{p-1} + \cdots + X+1$

Can someone give me a hint how to the irreducibility of $X^{p-1} + \cdots + X+1$, where $p$ is a prime, in $\mathbb{Z}[X]$ ? Our professor gave us already one, namely to substitute $X$ with ...
2
votes
2answers
207 views

About Lipschitz condition on polynomial mapping from $\mathbb R$ to $\mathbb R$

Let $p$ be a polynomial. Show that $p\colon\Bbb R \to \Bbb R$ is Lipschitz iff the degree of $p$ is less than $2$.
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0answers
365 views

Analytic solution for a quartic polynomial

I'm trying to use a generic solution with an analytic method for quartic polynomials. I am only interested in finding the real roots. Based on the solutions given by Wolfram|Alpha, and since the ...
1
vote
3answers
58 views

Find pair of polynomials a(x) and b(x)

If $a(x) + b(x) = x^6-1$ and $\gcd(a(x),b(x))=x+1$ then find a pair of polynomials of $a(x)$,$b(x)$. Prove or disprove, if there exists more than 1 more distinct values of the polynomials.
0
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1answer
63 views

Find all $p\in\mathbb{R}[x]$ such that $p\circ q=q\circ p$ for all $q\in\mathbb{R}[x]$

Find all polynomials $p(x)$ such that $p(q(x)) = q(p(x))$ for every polynomial $q(x)$. Thanks
2
votes
1answer
512 views

Generator polynomial

I need to compute a generator polynomial for a binary cyclic code of length 12 and dimension 5. I know that factorization of $(x^{12}+1)$ over $GF(2)$ is $(x+1)^4(x^2+x+1)^4$. What will be next step? ...
2
votes
1answer
115 views

If $f$ is at least degree $1$, then $f(n)$ cannot be prime for all n

I can't figure it out. Can you give me some advice? Let $f$ be a polynomial in $\mathbb{Z}[X]$ of degree at least 1. Prove that $f(n)$ cannot be a prime for each $n \in \mathbb{Z}$. I tried induction ...
0
votes
1answer
59 views

Polynomial $P\in\mathbb{R}[x]$ with $\overline{P(\overline{x})}=P(x),\forall x\in\mathbb{C}$

I have already shown that any polynomial $P\in\mathbb{R}[x]$ satisfies $\overline{P(\overline{x})}=P(x),\forall x\in\mathbb{C}$ My question is, given a polynnomial $P\in\mathbb{C}[x]$, how I can ...
7
votes
3answers
159 views

Square of polynomial

I have a problem with the following task. Let $W(n) := an^2 + bn + c$ where $a,b,c \in \mathbb{Z}$. Assume that for all $n \in \mathbb Z$ we have that $W(n)$ is the square of an integer. Show that ...
0
votes
2answers
163 views

Polynomial sequence

Let sequence $(P_n), (Q_n)$ be defined as $(1),(2),(3)$. $(1)$ $P_{0} = Q_{0} = 1$ $(2)$ $P_n(x) = P_{n-1}(x) + x^{2^{n-1}} Q_{n-1}(x)$ $(3)$ $Q_n(x) = P_{n-1}(x) - x^{2^{n-1}} Q_{n-1}(x)$ Let k ...
2
votes
1answer
234 views

Multivariable Gauss's Lemma

Gauss's Lemma for polynomials claims that a non-constant polynomial in $\mathbb{Z}[X]$ is irreducible in $\mathbb{Z}[X]$ if and only if it is both irreducible in $\mathbb{Q}[X]$ and primitive in ...
1
vote
1answer
77 views

expanding $(a+b+c+…)^x$ where $x$ is natural neatly

Suppose that $a,b,c,d,...$ are unknown variables. One wishes to expand $(a+b+c..)^x$ where $x$ is natural number ina neat manner (for e.g. using combination, sigma etc.). What would be some way? ...
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0answers
63 views

How to approximate $\sqrt[3]{x}$ when $x$ is rational number

One wants to approximate the real value of $\sqrt[3]{x}$ when $x$ is rational number. One want to approximate to two decimal digits. Is there any way to do this quickly?
1
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1answer
163 views

A way of finding the root of the polynomial equation

What would a fast way to find the root (where equation becomes 0) of the polynomial equation? We all know the easy way to find the solution for quadratic equation, but not for others.. Would this ...
0
votes
3answers
72 views

Conjugation of polynomial in $\mathbb{Z}[x]$.

Let $a,b \in \mathbb{Q}$ and $d \neq 0,1$ be a square free integer. Define $\overline{a + b\sqrt{d}} = a - b\sqrt{d}$ If $f \in \mathbb{Z}[x]$ show that: $f(\overline{\alpha}) = ...
2
votes
3answers
199 views

Precision with Taylor Expansions

when you take a 1st order taylor expansion of a function, so: $$f(a) + f'(a)(x-a)$$ does that mean that if the result is only accurate to one decimal place? so for a value a.bcd, d would be the ...
0
votes
2answers
126 views

Quartic (degree 4) polynomial complex number problem

Can you find a quartic (degree 4) polynomial $p(x) = ax^4+bx^3+cx^2+dx+e$ with real coefficients $a$, $b$, $c$, $d$, $e$ whose roots are precisely $x=5$, $x=-2$, $x=3$ and $x=1+i$ ? Guys please help ...
11
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2answers
216 views

Is a bivariate function that is a polynomial function with respect to each variable necessarily a bivariate polynomial?

Let $ \mathbb{F} $ be an uncountable field. Suppose that $ f: \mathbb{F}^{2} \rightarrow \mathbb{F} $ satisfies the following two properties: For each $ x \in \mathbb{F} $, the function $ ...
1
vote
3answers
165 views

Reducible polynomial + integer = Reducible polynomial?

Reducible polynomial + integer = Reducible polynomial ? As the title says. More specific : For every integer $n$, does there exist a pair of polynomials $p(x)$ and $q(x)$ such that: ...
0
votes
2answers
65 views

Irreducibility of $x^{3}-t\in\mathbb{C}(t)[x]$

Denote $F=\mathbb{C}(t)$ and consider $p(x)=x^{3}-t\in F[x]$ Is it true that $p$ is irreducible over $F$ ? My thoughts: I think that since it is not true that $t^{2}\mid t$ (I don't know how to ...
0
votes
2answers
501 views

show two polynomials interpolate the same data set

I need some hint on this particular homework problem. Show the following two polynomials $R(x)$ and $L(x)$ both interpolate the given points $$\left\{(x_1,\ y_1),\ (x_2,\ y_2),\ (x_3,\ y_3),\ ...
5
votes
4answers
7k views

Inverse function of a polynomial

What is the inverse function of $f(x) = x^5 + 2x^3 + x - 1?$ I have no idea how to find the inverse of a polynomial, so I would greatly appreciate it if someone could show me the steps to solving this ...
1
vote
1answer
373 views

Using Lagrange's Interpolation Formula to show that boolean functions over finite fields are polynomials

Let $F_2$ be the set of all the functions from the finite field $GF(2^n)$ of $2^n$ elements to $GF(2)$. I am reading a textbook that proves that the elements of $F_2$ can be represented by ...