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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1answer
134 views

Find a degree 5 polynomial $f \in \mathbb Z_5[x]$ with exactly 4 distinct roots

Find a degree 5 polynomial $f \in \mathbb Z_5[x]$ so that it has exactly 4 distinct roots and factorize it as product of irreducible factors. I'm really struggling in finding such polynomial, so ...
1
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1answer
102 views

Factorize $f$ as product of irreducible factors in $\mathbb Z_5$

Let $f = 3x^3+2x^2+2x+3$, factorize $f$ as product of irreducible factors in $\mathbb Z_5$. First thing I've used the polynomial reminder theorem so to make the first factorization: ...
2
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3answers
126 views

question related to radical sign

My question is- Let $p(x)= \sqrt{x + 2 + 3\sqrt{2x-5}} - \sqrt{x - 2 + \sqrt{2x-5}}$. Then $p(2012)^6$ equals? Any solution for this question would be greatly appreciated. Thank you,
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2answers
72 views

Find monic grade 3 polynomial in $\mathbb Z_p[x]$ then factorize

Let $f = 15x^4+22x^3-x=0$ a polynomial in $\mathbb Z_p[x]$, find the first prime $p$ value that will make $f$ result in being grade 3 and monic. Then factorize $f$ in $\mathbb Z_3[x]$ as product of ...
0
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1answer
65 views

How to simplify $(3a-b^2-a)^3$ by using special product?

When I simplify $(3a-b^2-a)^3$, I used $(u±v)^3=u^3±3u^2v+3uv^2±v^3$ but I'm confused with $(b^2-a)$
2
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2answers
57 views

Sum of the thirteenth power of the roots of given polynomial

Find the sum of the thirteenth powers of the roots of $x^{13} + x - 2\geq 0$. Any solution for this question would be greatly appreciated.
2
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3answers
104 views

Theorem for Dividing Polynomials

When a polynomial $$P(x)=x^4- 6x^3 +16x^2 -25x + 10$$ is divided by another polynomial $$Q(x)=x^2 - 2x +k,$$ then the remainder is $$x+a.$$ I have to find the values of $a$ and $k$. Can ...
4
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1answer
120 views

Is $(x^3-x^2+2x-1)$ prime in $\mathbb{Z}/(3)[x]$?

This is somewhat of a follow up on this question: Why is $(3,x^3-x^2+2x-1)$ not principal in $\mathbb{Z}[x]$? I'm curious, is $\mathbb{Z}[x]/I$ a domain, with $I=(3,x^3-x^2+2x-1)$? I know $I$ is not ...
0
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2answers
281 views

Why can/do we multiply all terms of a divisor with polynomial long division?

I'm trying to understand why polynomial long division works and I've hit a wall when trying to understand why we multiply all terms of the divisor by the partial quotient. Consider: $$\frac{x^2 + 3x ...
0
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2answers
83 views

Find a prime number $p$ so that $f = \overline{3}x^3+ \overline{2}x^2 - \overline{5}x + \overline{1}$ is divided by $x-\overline{2}$ in $\mathbb Z_p$

Let $f = \overline{3}x^3+ \overline{2}x^2 - \overline{5}x + \overline{1}$ be defined in $\mathbb Z_p$. Find a prime number $p$ so that $f$ can be divided by $g = x-\overline{2}$, then factorize $f$ as ...
0
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1answer
105 views

quadratic polynomial investigation

in my mathematics textbook,i have found one interesting problem and i have one question.textbook asks following problem deduce all possible value of $a$,for which equation $4*x^2-2*x+a=0$ has ...
7
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4answers
335 views

Nth derivative of $\tan^m x$

$m$ is positive integer, $n$ is non-negative integer. $$f_n(x)=\frac {d^n}{dx^n} (\tan ^m(x))$$ $P_n(x)=f_n(\arctan(x))$ I would like to find the polynomials that are defined as above ...
3
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1answer
227 views

Proof of Factor Theorem

I'm following the proof of the Factor theorem in here but I don't understand how $\deg(X-r) = 1$.
3
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1answer
221 views

Asymptotics of an expression of the root of a polynomial

Given that $x_0$ is the unique positive solution of $(2-x)^{n+1}=x(x+1)\cdots(x+n)$, try to find the asymptotic value of $$ M=\prod_{k=0}^n\left(\frac{k+2}{k+x_0}\right)^{k+2} $$ with absolute error ...
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3answers
113 views

How to see that the shift $x \mapsto (x-c)$ is an automorphism of $R[x]$?

In the process of studying irreducibility of polynomials, I encountered the criterion that $p(x)$ is irreducible if and only if $p(x-c)$ is irreducible. When trying to determine what properties of the ...
5
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1answer
767 views

Can any Polynomial be factored into the product of Linear expressions?

Specifically I am wondering if... Given a Polynomial of n degree in one variable with coefficients from the Reals. Will every Polynomial of this form be able to be factored into a product of n ...
3
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2answers
63 views

Determine $a$ values allowing $x^2+ax+2$ to be divided by $x-3$ in $\mathbb Z_5$

Determine for which $a$ values $f = x^2+ax+2$ can be divided by $g= x-3$ in $\mathbb Z_5$. I don't know if there are more effective (and certainly right) ways to solve this problem, I assume there ...
0
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3answers
193 views

Determine monic and degree 3 polynomial in $\mathbb Z_p$

I stumbled upon this kind of problem and I really can't get the hang of it. Will anyone please outline the way to solve it? Determine for which of the first $p > 0$ values the polynomial $f = ...
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3answers
256 views

Why is $(3,x^3-x^2+2x-1)$ not principal in $\mathbb{Z}[x]$?

I have a small hitch in showing $(3,x^3-x^2+2x-1)$ is not principal in $\mathbb{Z}[x]$. Towards the contrary, I suppose $(3,x^3-x^2+2x-1):=(3,f)=(g)$ is principal. Then $3\in (g)$, so $3=gh$ for some ...
2
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4answers
85 views

Turn fractions into $\mathbb Z_7$ elements

I had to perform a division between two polinomials $2x^2+3x+4$ and $3x+4$, my book suggests to do this operation without worrying about the modulo. So my result is ...
2
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4answers
190 views

Proving all roots of a sequence of polynomials are real

Let the sequence of polyominoes $R_n(z)$ be defined as follows for $n\geqslant1$: $$R_n(z)\;= \;\sum_{r=0}^{\lfloor\frac{n-1}{2}\rfloor} \tbinom{n}{2r+1}(4z)^r.$$ I would like to prove that all the ...
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246 views

When is $\mathbb{F}_p[x]/(x^2-2)\simeq\mathbb{F}_p[x]/(x^2-3)$ for small primes?

I've been considering the rings $R_1=\mathbb{F}_p[x]/(x^2-2)$ and $R_2=\mathbb{F}_p[x]/(x^2-3)$, where $\mathbb{F}_p=\mathbb{Z}/(p)$. I'm trying to figure out if they're isomorphic (as rings I ...
2
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3answers
108 views

Is there a good way to solve for the inverse of $(u^2-u+4)$?

I'm having trouble calculating the inverse of a polynomial. Consider the polynomial $f(x)=x^3+3x-2$, which is irreducible over $\mathbb{Q}$, as it has no rational roots. So $\mathbb{Q}[x]/(f(x))\simeq ...
5
votes
2answers
299 views

Polynomials representing primes

Suppose over $\mathbb{Z}$ we are given an irreducible polynomial $p(x)$. Can we say that $p(x)$ at least represents a prime as $x$ runs through integers? Thanks in advance
0
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1answer
59 views

Polynomials with roots having the same module and linear dependent arguments

Is it possible for a polynomial with integer coefficients to have some of its roots: $$m_1e^{i\theta_1 \pi}, m_2e^{i\theta_2 \pi}, \ldots, m_ke^{i\theta_k \pi}$$ such that there exist nonzero integers ...
2
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3answers
8k views

Given four points on a cubic function curve, how can I find the curve's function?

Say I have a curve $$y = ax^3 + bx^2 + cx + d.$$ I don't know $a$, $b$, $c$ or $d$, but I do know the $(x,y)$ values of four points on this curve. How can the values of $a$, $b$, $c$ and $d$ be ...
3
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3answers
103 views

Unique expression of a polynomial under quotient mapping?

I have a weird feeling about something I'm reading. Suppose $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ is a polynomial over a field $F$. Let $y=x+(f(x))$ be the image of $x$ in the quotient ...
0
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1answer
158 views

Maximal number of monomials of multivariate polynomials.

If we consider multivariate polynomials of order $N$ and $n$ variables, how do we show that for the maximal number of monomials the following holds: $\binom{N+n}n$. The hint our teacher gave was that ...
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1answer
165 views

How to factor $x^5 - x + 1$

As I understand it $x^5 - x + 1$ is not solvable by radicals. But it splits over $\mathbb{C}$, so how does it factor into linear factors?
4
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1answer
131 views

Is the free commutative monoid on $n$ generators essentially the same as a polynomial ring in $n$ variables?

I let $M$ denote the free commutative monoid generated by some elements $x_1,\dots, x_n$. Suppose too that $R$ is a commutative ring. Then I can construct the free algebra of $M$ over $R$, denote it ...
2
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1answer
50 views

Is there an explicit way to determine $\mathrm{Mat}_n(R[X_1,\dots,X_m])\simeq\mathrm{Mat}_n(R)[X_1,\dots,X_m]$?

For a commutative ring $R$, let $\mathrm{Mat}_n(R[X_1,\dots,X_m])$ denotes the matrix ring with entries from $R[X_1,\dots,X_m]$, and let $\mathrm{Mat}_n(R)[X_1,\dots,X_m]$ denotes the polynomial ring ...
2
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1answer
162 views

Is there a unique homomorphism extension for noncommutative rings in this proof?

I read the following in my book: Let $R$ and $S$ be commutative rings, $\eta$ a homomorphism of $R$ into $S$, $u$ an element of $S$. Let $R[x]$ be the ring of polynomials over $R$ in the ...
2
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0answers
122 views

Coefficients of a product of polynomials of the form $1+x+\cdots+x^k$

I'm looking for the coefficients $a_0,\ldots,a_k$ of the polynomial $$f(x)=\prod_{i=1}^r(1+x+\cdots +x^{k_i-1})=\prod_{i=1}^r\frac{1-x^{k_i}}{1-x}$$ Since $f(1/x)=x^{-k}f(x)$ where $k = ...
2
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2answers
129 views

Polynomial-related manipulation

My question is: Factorize: $$x^{11} + x^{10} + x^9 + \cdots + x + 1$$ Any help to solve this question would be greatly appreciated.
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1answer
110 views

Polynomials in 2 variables

For all complex polynomials in 2 variables I know, the set of zeros looks like a union of curves. Wrong: I can get a circle with $x^2+y^2-1$ or two vertical lines with $(x-1)(x-2)$?. Can I get ...
1
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2answers
145 views

Polynomial over characteristic two finite field of odd degree with certain image

I am trying to construct a polynomial $f \in \mathbb{F}_{2^k}$ of odd degree, such that $\forall x \in \mathbb{F}_{2^k} \exists \alpha \in \mathbb{F}_{2^k}$ such that $f(x)=\alpha ^2 -\alpha$. ...
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2answers
71 views

Is there a concrete description of the ideal $I$ such that $\mathbb{Q}[x]/I\cong\mathbb{Q}[\sqrt{2}+\sqrt{3}]$?

I know that in general if $R[u]$ is the ring obtained by adjoining an element $u$ to a ring $R$, then $R[u]\cong R[x]/I$ for some ideal $I$ such that $I\cap R=\{0\}$. In a particular instance, I'm ...
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1answer
377 views

For a trigonometric polynomial $P$, can $\lim \limits_{n \to \infty} P(n^2) = 0$ without $P(n^2) = 0$?

Disclaimer: The original version of this question focused on $2^n$ in lieu of $n^2$. It is in the hope that the question is easier with $n^2$ that I changed it. I have an always-nonnegative (on the ...
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1answer
234 views

maximum of the function at limit

I have a simple question. Let $$P(\theta;K) = \left(1-\theta\right)^K\left[\frac{1-(1-\theta)^K-\theta^K}{(1-\theta)^K+\theta^K}-\sum_{i=1}^{\frac{K-1}{2}}\left(\begin{array}{l} K \\ ...
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1answer
93 views

Question about polynomials and congruences

Let be $m(X), n(X), a(X), b(X), m(X), n(X), g(X) \in F[X]$ a polynomials, where $F$ is a finite field with characteristic 2. Let be a relations $a \equiv b \mod g(X)$ and $m \equiv n \mod g(X)$. Are ...
3
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1answer
252 views

Is this polynomial positive?

Let $p\geq 2$, and $p$ is not a half odd integer. $t\in R$. Is the following polynomial positive: $$ T_k(t)=\left(\frac ...
34
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1answer
1k views

Decomposing polynomials with integer coefficients

Can every quadratic with integer coefficients be written as a sum of two polynomials with integer roots? (Any constant $k \in \mathbb{Z}$, including $0$, is also allowed as a term for simplicity's ...
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1answer
176 views

Homogeneous polynomials in two variables taking integer values

It is known that ${x\choose 0},{x\choose 1},\ldots,{x\choose n}\in\mathbb{Q}[x]$ is a $\mathbb{Z}$-basis for set of polynomials of degree at most $n$ which map $\mathbb{Z}$ into itself. For fixed ...
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138 views

A special factorization

Suppose that monic polynomial $f(x)\in\Bbb Z[x]$ such that for all $m\in\Bbb Z$, $m>1$, there's no integers $\langle r,r_1,\ldots,r_m\rangle$ such that $f(r)=f(r_1)\cdots f(r_m)$. Is there any ...
2
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2answers
59 views

How smooth is the distribution function of a convex polynomial?

Here is a prototype of the problem I have in mind: Let $P:\mathbb{R}^2\rightarrow\mathbb{R}$ be a strictly convex, nonnegative polynomial such that $P(0,0)=0$. Let $\alpha\geq 0$, and consider the ...
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0answers
132 views

optimization of polynomial function with linear constraints

I have a polynomial function, which is actually the Cobb-Douglas production function, of the form $f(x,y) = \frac{\{x^{\alpha} y^{1-\alpha}\}^{1-\gamma}}{1-\gamma}$ with linear constraint K(x,y)= ...
0
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1answer
79 views

Restoring a point after transformation

I am given a point $ \begin{bmatrix} u & v \end{bmatrix}^T $ which I know is in form $\begin{bmatrix} \frac{x}{f(r)} & \frac{y}{f(r)} \end{bmatrix}^T$ where $f(r)$ is polynomial function, ...
1
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0answers
127 views

Curvature of cubic 3D curve

I want to calculate the curvature of a surface defined by a set of $(x,y,z)$ coordinates. So I fitted this formula to the set of points and obtained values of $a, \dots,k$ with $R^2 \sim 0.98$: $$z = ...
5
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0answers
507 views

Gauss-Lucas Theorem (roots of derivatives)

Gauss-Lucas Theorem states: "Let f be a polynomial and $f'$ the derivative of $f$. Then the theorem states that the $n-1$ roots of $f'$ all lie within the convex hull of the $n$ roots ...
3
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0answers
120 views

Extension of the theorem of Jacobson

Let $A$ be a ring. Let $E$ be the set of polynomials $\{X^n-X \in \mathbb{Z}[X]|n \in \mathbb{N}^*-\{1\}\}$. By the theorem of Jacobson, we know that if for each $a\in A$ there is an element of $E$ ...