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
13 views

Smallest dimension for having two non similar matrices with same minimal and characteristic polynomials

I give here the example of two non similar matrices, namely $$M=\begin{pmatrix} 1 & 1 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1 ...
0
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2answers
73 views

$x^4 + 4 y^4$ never a prime $>5$?

Let $x,y$ be nonzero integers. I could not find primes apart from $5$ of the form $x^4 + 4 y^4$. Why is that ? I know that if x and y are both not multiples of $5$ then it follows from fermat's ...
8
votes
1answer
97 views

Primitive polynomials $P$ with $\gcd(P(x),P(y))=1$ for infinitely many $x,y$

Characterize all primitive polynomials $P$ having integer coefficients such that there exist infinitely many natural numbers $x,y$ with $\gcd(P(x),P(y))=1$ NOTE: A primitive polynomial is ...
16
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1answer
145 views

Prove this deg inequality $\deg{P(x)}\cdot \deg{Q(x)}\cdot \deg{R(x)}\ge 656$

Let three non-constant polynomails $P(x),Q(x),R(x)\in \mathbb Z[x]$,and if this equation $P(x)Q(x)R(x)=2015$ has $49$ distrinct integer roots. Prove that $$\deg{P(x)}\cdot \deg{Q(x)}\cdot ...
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1answer
36 views

Finding values of $a, b$ such that $0\le x^4 +x^3 +ax+b\le (x^2-1)^2$

Given real values of $a, b$ such that for all $x\ge0$, $$0\le x^5+x^3+ax+b\le (x^2-1)^2\ ,$$ find the value of $ab$. What I've done is let $x=1$, thus $$0\le2+a+b\le0$$ this forces $a+b=-2$. let ...
5
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1answer
36 views

Closed formulas for two Poincaré series

Associated with an arbitrary direct sum $E = \bigoplus_{i \ge 0} E_i$, of finite dimensional $k$-vector spaces $E_i$, $i = 0, 1, 2, \dots,$ there is a formal power series $P_E$, with nonnegative ...
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0answers
19 views

Curve fitting within bounded domain

Basically I have a function and I can get the value (without noise) of the function but I don't know the equation. I want a perfect (or near perfect) fit for the function just within certain domain. ...
9
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2answers
79 views

Exists polynomial satisfying following?

Let $s, u \in M_m(\mathbb{k})$ be a pair of commuting matrices such that $s$ is a diagonal matrix and $u$ is a strictly triangular matrix (with zeros on the diagonal). Put $a = s + u$. Does there ...
3
votes
1answer
43 views

Chidzalo's Sequence

One day I observed that if $ \displaystyle f(x)= \sum _{i=0} ^{n} k _{i} x ^{i}$ is divided by the linear polynomial $ px-q $ where $p$ and $ q $ are constants, then the quotient is $\displaystyle ...
1
vote
1answer
28 views

factorisation of polynomial fraction

we have the fraction $\dfrac{x^2}{x^2-2x+1}$ (1) we can easily factorize the denomirator by using the polynomial's roots $\dfrac{x^2}{(x-1)^2}$ (2) now I saw somewhere that you can even simplify ...
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1answer
14 views

Differentiation of Rodrigues' formula

Iam trying to differentiate Rodrigues' formula m times with respect to x. to attain the form Any help ?
1
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1answer
31 views

Maximal ideals of a quotient of a polynomial ring over an algebraically closed field

Suppose that $k$ is an algebraically closed field. Then, we know from Hilbert's (weak) Nullstellensatz that the maximal ideals of $k[x_1, \ldots, x_n]$ are of the form $(x_1 - a_1, \ldots, x_n - a_n)$ ...
1
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1answer
33 views

Finding all possible cubic equations from two/three points

I'm trying to find all possible cubic equations that can be found from two scenarios. The first scenario is a lot like the one I asked a couple of days ago on Stack Overflow, found here: ...
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4answers
142 views

Finding roots of the polynomial $x^4+x^3+x^2+x+1$

In general, how could one find the roots of a polynomial like $x^4+x^3+x^2+x^1+1$? I need to find the complex roots of this polynomial and show that $\mathbb{Q (\omega)}$ is its splitting field, but I ...
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0answers
26 views

Question about a certain type of polynomials

Let $p:\mathbb{C}\rightarrow\mathbb{C}$ be a complex polynomial of degree $n\in\mathbb{N}$ and that $\text{Re }p(z)\geq0$ when $z\in\mathbb{R}$. Moreover impose that $p(z)$ has a point ...
1
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0answers
35 views

divisibility of $a^m+a-1$ by $a^n+a^2-1$.

Find all integers $m,n\geq 3$ such as the polynomial $a^m+a-1$ is divisible by $a^n+a^2-1$. It is clear that $m=n+k$ for some integer $k\geq 0$, we find that ...
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0answers
19 views

Plot zeroes of polyomial in $\mathbb{R}[x,y,z]$ in Linux

I have a polynomial in $\mathbb{R}[x,y,z]$, say $$p_c(x,y,z)=x^2+y^2+z^2-xyz-c.$$ Do you know any simple piece of software to plot the zeroes of this polynomial using Linux? In the past I plotted ...
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2answers
42 views

Making a Perfect square [closed]

How to get max value of $-2x^2+3x+5$ by making perfect square. I got wrong ans. Correct ans. Must be $49/8$.
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1answer
25 views

Quadratic problems [closed]

For the equation $f(x)=x^5+ax^4+bx^3+cx^2+dx-420$, why do we use $f(x)=(x-\alpha)(x-\beta)(x-\gamma)(x-\delta)(x-\mu)$ with multiplication of the roots giving$\alpha\beta\gamma\delta\mu=420= 2\times ...
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1answer
30 views

Derivative of polynomial in GF(16)

how can I find the derivative of the following polynomial in $GF(2^4)$: $\alpha x^4+x^3+\alpha x^2+\alpha^2 x+1$ ?
6
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1answer
174 views

The asymptotic of the number of integers that are sums of three nonnegative cubes

Let $c(n) $ be the number of distinct integers between $0 $ and $n $ of the form $ a^3 + b^3 + c^3$, meaning the sum of $3$ nonnegative cubes. $C(n) = O( n \space \ln(n)^x ) $ Find and prove the ...
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0answers
30 views

Integral roots of a polynomial

I have one doubt. Suppose, $f_{n}(x)=a_0x^n+a_1x^{n-1}+a_2x^{n-2}+,...,+a_{n-1}x+a_n=0$ be a polynomial with an integral coefficients. If for some $n$ ( say $n=2 \ or \ 3$) , $f_{n}(t)=0,$ where, $t ...
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1answer
29 views

Find all polynomials $P \in \mathbb{R}[x]$

Find all polynomials $P \in \mathbb{R}[x]$ such that $$P(x^2-2x)=P(x-2)^2.$$ I think we should replace $x$ with $x+1$, I really don't know, any help?
0
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1answer
46 views

Finding real roots of a Polynomial Equation without graphs.

I am interested in finding the number of real roots of this polynomial : $x^9 + \frac{9}{8}x^6 + \frac{27}{64}x^3 - x + \frac{219}{512} = 0$ Okay, I know that graphing it would tell me how many real ...
3
votes
3answers
45 views

Conditions for distinct real roots of cubic polynomials.

Given a cubic polynomial with real coefficients of the form $f(x) = Ax^3 + Bx^2 + Cx + D$ $(A \neq 0)$ I am trying to determine what the necessary conditions of the coefficients are so that $f(x)$ has ...
2
votes
2answers
58 views

a matrix of rank $r$ satisfies a polynomial of degree $r+1$.

Let $M$ be an $n\times n$ matrix with coefficients in $\mathbb C$. Suppose $M$ has rank $r$ with $r<n$. Prove there is a polynomial $P(x)$ with degree $r+1$ and coefficients in $\mathbb C$ such ...
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votes
2answers
52 views

How do I find lowest upper bound and greatest lower bound when dealing with functions? [closed]

Here is my problem: I have to find the integer that is the highest lower bound for the roots of $$f(x)=x^4-3x^2+2x-4$$ I am not sure how to do this and the book I am using does not explain it very ...
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2answers
98 views

Grade 7: When is an Algebraic Expression NOT a Polynomial? [duplicate]

Ok so this appeared in our school, and we all didnt know the answer. We arent sure, but we think the answer is, "when there are Signs like the Square Root signs, Negative Integers, etc." Or "When a ...
3
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1answer
59 views

Non-linear equivariant maps between group representations

Given two representations $\pi_1$ and $\pi_2$ of a group $G$ (let's say it's a compact Lie group), a natural thing to study are linear equivariant maps A between them: $$ A \pi_1 = \pi_2 A $$ I'm ...
2
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4answers
70 views

Simplifying $\sum_{i=1}^{n-2}i(n-1-i)$

I have been trying to simplify $\sum_{i=1}^{n-2}i(n-1-i)$ i.e - remove the summation, put it in polynomial form Since $i$ is the changing variable, I don't think this is possible. I also know that ...
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2answers
87 views

Need help with an Elementary Math question [closed]

If $a+b+c=1$ and $ax^2 + bx + c = 0$ has a unique solution. Find $a,b$, and $c$.
1
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1answer
36 views

semi definite representation of a polynomial

Let $P\left( x \right) = {a_0} + {a_1}x + ..... + {a_n}{x^n}$ be a polynomial with degree $n$. How can I write the vector ${\bf{a}} = [{a_0},....{a_n}]$ in a semidefinite matrix form which yields ...
0
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1answer
12 views

Offsetting a 2-D polynomial

I have a surface that is defined using a two dimensional polynomial: $$z = f(x) + g(y)$$ I want to offset the curve in the $XY$ plane from a point on the surface $\left(x_0, y_0, z_0\right)$ to a ...
0
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2answers
48 views

Find the number of distinct integer roots of $P^2 (x)-1$

Let $P(x)$ be a polynomial with integer coefficients of degree $d>0$. Prove that the number of distinct integer roots of $P^2(x)-1$ is at most $d+2$.
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votes
2answers
83 views

How can I use left inverse to f(x)=3x format equation? [closed]

I want to solve linear equations as following. $$f(x)= 3x^3 -4x^2 +3x -7$$ $$f(x)= 2x^3 -3x^2 +2x -1$$ $$f(x)= 1x^3 -7x^2 +1x -2$$ But these seem that there is no $y$. How can I solve by using ...
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votes
1answer
19 views

Scale Proportionally a Polynomial [closed]

I have 3 polynomial functions that all overlap (to form a shape): I need to scale the polynomials down to half the current size. How do I do this? It must be scaled down proportionally ($x$ and ...
0
votes
1answer
25 views

Linear map with polynomials - Find a matrix

Let $F:P_3\to P_3$ be a linear map given by $F(p(x))=(x+1)p'(x)$ (where $p'(x)$ denotes the derivative). (i) Specify the matrix $A$ for $F$ in the basis $\{1,x,x^2,x^3\}$ (ii) Determine a basis of ...
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5answers
53 views

Why is discriminant less than zero?

the question is find the range of values of $c$ for which the expression $4x^2-4x+4c^2-8$ is non-negative for all real values of $x$. I got the discriminant but I don't understand why it has to be ...
5
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2answers
44 views

Determining the minimal polynomial over $\Bbb{Q}$

I was working on a homework assignment from Hungerford: Find the minimal polynomial of the element $\sqrt{1+\sqrt{5}}$ over $\Bbb{Q}$. Naturally the solution would be the polynomial with roots ...
0
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1answer
38 views

What are some applications of “separable” spaces?

A separable space is a space that contains a countable dense subset. For example, the space of continuous functions $C[a,b]$ is separable. Are there some practical applications arising out of this ...
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3answers
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$\sum_{j=0}^{n-1}z_j^k=\begin{cases} 0, & \text{if $1\leq k \leq n-1$ } \\ n, & \text{if $k=n$ } \end{cases}$

Show that $\sum_{j=0}^{n-1}z_j^k=\begin{cases} 0, & \text{if $1\leq k \leq n-1$ } \\ n, & \text{if $k=n$ } \end{cases}$, where $z_0,...,z_{n-1}$ are the $n$-th roots of unity. For $k=n$ it ...
0
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0answers
12 views

does this have a unit modulus solution?

Let $z_m, m=0,1,2..N-1$ be arbitrary complex numbers and form the polynomial product in the variable $\rho$: $P(\rho) = \left [\sum_{m=0}^{N-1} (m \bar z_m \rho^m) \right ] \left ...
0
votes
1answer
45 views

Find out whether a polynomial is irreducible or not

Let $f=X^7-(7-6i)X^3+5X^2+3+6i\in\mathbb{Z}[i][X]$. Check whether $f$ is irreducible: over $\mathbb{Z}[i]$ over $\mathbb{Q}(i)$ Probably I will have to use Einstein criterion with some ...
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0answers
20 views

Proof of determinant formula and coprime polynomial

Problem: Let $p(z)=p_o+p_1z+...+p_{n-1}z^{n-1}$ be a polynomial of maximum degree $n-1$. Show that $p(z)$ and $z^n-1$ are coprime if and only if $$\begin{vmatrix} p_0 & p_{n-1} & ... & ...
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0answers
18 views

Primitive elements of polynomial

Problem: Given a primitive polynomial $f(x)$ over $\mathbb F_2$ of degree 248, a) is $g(x) = x^{17}$ a primitive element? Why? b) is $h(x) = x^{23}$ a primitive element? Why? Progress: I'm ...
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0answers
21 views

Explanation of symmetric sum in a solution

Can someone explain me why $x+y=5$ in $\text{E8}$ clearly.
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0answers
35 views

How can I write $\prod\limits_{i = 0}^n {\left( {x - {x_i}} \right)} $ in terms of a polynomial as $\sum\limits_{i = 1}^n {{a_i}{x^n}} $?

How can I write $\prod\limits_{i = 0}^n {\left( {x - {x_i}} \right)} $ in terms of a polynomial as $\sum\limits_{i = 1}^n {{a_i}{x^n}} $? In the other words, is there a way to write $a_i$ in terms of ...
2
votes
1answer
9 views

Partial Derivative of the root of a polynomial with respect to its coefficients

Suppose that I have a polynomial $p(x) = c_0+c_1x^{-1}+\ldots c_{N-1} x^{-(N-1)}$ with roots $x_0,\ldots,x_{N-1}$, how do I derive an expression for the partial derivative of the root with respect to ...
3
votes
3answers
268 views

Algebra question from practice GRE exam

The following is a question from the GRE exam GR9367: Let $n > 1$ be an integer. Which of the following conditions guarantee that the equation $x^n = \sum_{i=0}^{n-1} a_ix^i$ has at least one root ...
1
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2answers
28 views

Factoring a polynomial with complex coefficients

Given $$3z^2+6z+3i=0$$ Find the complex roots and write in the form $a+bi$. I want to see how to factor it when there is an $i$ being multiplied by the constant.