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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269 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 ...
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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.
2
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0answers
27 views

Intuition/Derivation for Newton's Sums?

I often come across problems in which I want to find the sum of the $k$'th powers of the roots of a polynomial. I have heard of a method known as the Newton-Girardae formulae. However, I cannot ...
3
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0answers
60 views

Finding exact roots

I know of the rational root theorem to find all rational zeros and Newtons method of approximating zeros, but what if all the solutions are irrational/imaginary and you need exact answers for the ...
2
votes
1answer
99 views

Quartic polynomial taking infinitely many square rational values?

I was wondering whether the value of $$P(x)=x^4-6x^3+9x^2-3x,$$ is a rational square for infinitely many rational values of $x$. Is there a general method to check this for a polynomial (in one ...
2
votes
2answers
67 views

For what values of $ a, b$ does the equation have real roots?

For what values of $a,b$ does the equation $${ x }^{ 2 }+2\left( 1+a \right) x+\left( 3{ a }^{ 2 }+4ab+4{ b }^{ 2 }+2 \right) = 0$$ have real roots? For it to have real roots, the ...
0
votes
1answer
47 views

Solving algebraic equations with radicals

I have several problems requiring assistance. Solve for x: $x\left( x-\sqrt { 3 } \right) \left( x+1 \right) +3-\sqrt { 3 } \quad =\quad 0$ I've followed the suggestion to get x^2 - (√3 -1)x + ...
0
votes
1answer
35 views

Showing irreducibility of a polynomial. [duplicate]

How would you go about showing that $p(x)=\frac{x^5-1}{x-1}=x^4+x^3+x^2+x+1$ is irreducible over $\mathbb{Q}$. I'm having trouble seeing how one can show whether this kind of polynomials are ...
0
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0answers
43 views

Eigenvalues of a matrix formed by derivatives of a polynomial

Let $f=f(x,y)$ be a polynomial with $x\geq 0,\ -2x\leq y\leq -x/2$ (so when $x=1, -2\leq y\leq-1/2$). Denote \begin{equation*} \begin{split} f_1=&\frac{\partial \ln f}{\partial x}(1,y),\\ ...
8
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0answers
211 views

How to simplify this combinatorial expression?

Find \begin{eqnarray} \sum_{j\in\mathbb{N}}(n-2j)^k\binom{n}{2j-m} \end{eqnarray} Note that this question is a generalization of this one. I tried to imitate the steps in the answer given in that ...
6
votes
1answer
78 views

Product of a Finite Number of Matrices Related to Roots of Unity

Does anyone have an idea how to prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} x^{-2j} & -x^{2j+1} \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} ...
1
vote
1answer
46 views

Prove that there is no integer k such that $f(k)=8$

Given, $f(x)=\sum_{i=0}^na_{i}x^{n-i}$ and $a_0=1$ have integral coefficients.If there exist four distinct integers a,b,c and d such that $f(a)=f(b)=f(c)=f(d)=5$,show that there is no integer $k$ such ...
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vote
4answers
80 views

If $\alpha$ and $\beta$ are the zeroes of $p(x) =x^2- px +q = 0$ [closed]

Find $\alpha^2 + \beta^2$ and $\alpha^3 + \beta^3$.
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0answers
28 views

Find the discriminant and comment on the discriminant.

Let $a_1,a_2,\dots,a_n$ be non zero real numbers and $b_1,b_2,\dots,b_n$ be real numbers.Find the discriminant of the quadratic equation ...
0
votes
1answer
25 views

Expand the equation and find the value of n. [closed]

If the equation \begin{equation}\sum_{i=1}^n(x+i-1)(x+i)=10n \end{equation} has roots $r$ and $r+1$, find $n$.
3
votes
1answer
48 views

Understanding the algebra of polynomials on a linear space

My advisor and I are working through a paper on partition functions, and we got to the following passage: Fix $n \in \mathbb N$ and let $W := ((\mathbb R^n)^{\otimes 3})^{C_3}$, where the $C_3$ ...
5
votes
3answers
106 views

Find $f(2015)$ given the values it attains at $k=0,1,2,\cdots,1007$

Let $f$ be a polynomial of degree $1007$ such that $f(k)=2^k$ for $k=0,1,2,\cdots, 1007$. Determine $f(2015)$. Taking $f(x)=\sum_{n=0}^{1007} a_n x^n $, (whence $a_0=1$), I tried to combine ...
3
votes
3answers
82 views

Factor $k^{4}+4k^{3}+8k^{2}+8k+4=0$ over $\mathbb C$

Any idea how to factor the polynomial $k^{4}+4k^{3}+8k^{2}+8k+4$ over $\mathbb C$? Candidates for rational roots are $\pm1, \pm2, \pm4$ but none of them satisfies $k^{4}+4k^{3}+8k^{2}+8k+4=0$.
0
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4answers
84 views

Find the remainder when $ p(x) = (x+2)^{101} + (x+3)^{200}$ is divided by $ x^2 +5x + 6 $. [closed]

Find the remainder when $p(x) = (x+2)^{101} + (x+3)^{200}$ is divided by $ x^2 +5x + 6 $.
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0answers
36 views

Vector-space of polynomial, find a basis of eigenvectors

Exercise: Let $P_3$ denote a vector space of polynomial of degree 3 or lower (and real coefficients) and let the linear map $F:P_4\to P_4$ be given by $F(p(t))=((t^2+1)p(t))^n$. Find a basis of ...
0
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1answer
15 views

finding the behavior of the asymptotes in a rational function

I'm having trouble understanding how to graph this function: $f(x) = \frac{x-2}{(x-4)(x+4)}$. The part I undertand: The x-intercept is (2,0) since x=2 makes the numerator zero. The y-intercept is ...
1
vote
2answers
65 views

Find $P(3)+8P\left ( \frac{3}{2} \right )$

If the polynomial $$P(x)=x^{3}+Ax^{2}+Bx+C$$ such that $P(1)=2 , P(2)=4$ Find $$P(3)+8P\left ( \frac{3}{2} \right )$$ A beautiful question, It's just for sharing a new ideas, thanks:)
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0answers
40 views

A Library that Finds a Polynomial's Roots Over a Field

I have a polynomial of high degree, whose coefficients are big integers. And the coefficients are elements of a prime order field. I want to find the roots of the polynomial over the field. I can use ...
6
votes
3answers
79 views

Prove : The polynomial has no integral roots. [duplicate]

Q. Prove that a polynomial $f(x)$,with integer coefficients has no integral roots if $f(0)$ and $f(1)$ are both odd integers. My attempt: Let $$f(x)=a_0+a_1x+a_2x^2+\dots+a_nx^n$$ now $f(0)=a_0$ ...
2
votes
1answer
37 views

I have plugged $p/q$ into the equation. Not sure what to do next.

Suppose $a_0,a_1,\dots,a_n$are integers and $a_0\neq 0$ and $a_n\neq 0$.Consider the polynomial $f(x)=a_0x^n+a_1x^{n-1}+\dots+a_{n-1}x+a_n$. If $p\neq 0,q>0$ are coprime integers and $p/q$ ...
3
votes
3answers
37 views

Unable to find the f(x)

Find the Cubic in $x$ which vanishes when $x=1$ and $x=-2$ and has values $4$ and $8$ when $x=-1$ and $x=2$ resprectively. I have proceeded like $P(x)=(x-1)(x+2)f(x)$ but I unable to find $f(x)$ ...
0
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0answers
13 views

Resizing LDPC Parity Matrices

I am having a hard time trying to understand different optimizations for LDPC codes. Most important, I am finding it hard to understand how to compare different codes, given their lengths. For ...
3
votes
0answers
44 views

Polynomial products [duplicate]

This problem $$ \large \displaystyle\prod \limits^{14}_{k=1}\cos \left( \frac{k \pi }{15} \right) =\ ? $$ I solved it in this way $$ x = \displaystyle \prod \limits^{14}_{k=1}\cos \left( \frac{k\pi ...
0
votes
1answer
38 views

How would I find the Field's elments generator?

Suppose we would construct a Field $F=GF(2^4)$ by using $f(x)=x^4+x^3+x^2+x+1$. In this case the generator is $\alpha=x+1$. Why $\alpha$ here is equal to $x+1$ how would I find this?
2
votes
1answer
30 views

How to calculate the polynomial coefficients of a fraction of polynomials?

I have a polynomial fraction which results in a polynomial $$\frac{f(x)}{g(x)}=q(x)$$ with $f$ $g$ and $q$ being polynomials. I have formulas for the coefficients of $f(x)$ and $g(x)$ dependent on the ...
4
votes
2answers
49 views

Is there any way to compute these sums quickly?

I have a sum of the following form (all numbers are positive integers): $$F(p) = \sum_{x=1}^{N} a_x x^p $$ Where $N$ and all $a_x$ terms are known/fixed constants. However I need to be able to ...
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votes
2answers
53 views

I don't understand why if two polynomials degrees do not exceed $n$ and they coincide at $n+1$ points then they are equal.

If polynomials P and Q has degrees not exceeding n and coincide at n+1 different points, then they are equal. $\text{Reference Below}$:
0
votes
1answer
28 views

Extracting variable from a couple of rational functions

Let $P$ and $Q$ be two rational functions of $z$ (coefficients over $\mathbb{C}$). How can one decide whether $z = R(P(z), Q(z))$ for some rational function $R$ of two variables, and if it is the case ...
2
votes
1answer
42 views

Representing as sum of squares of polynomials

Show that the polynomial $x^4y^2+y^4z^2+z^4x^2-3x^2y^2z^2$ cannot be written as the sum of squares of polynomials over $\mathbb{R}$ in $x, y, z$. I could not make any progress/significant ...
0
votes
0answers
27 views

Why do I get homogenizations of polynomials by trying to find roots in $\mathbb Q$.

I noticed that if I have a polynomial equation in, say $x$ that needs to be solved in $\mathbb Q$, one tactic is to substitute $x=y/z$ where $y$ and $z$ are coprime integers, but then after clearing ...
0
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0answers
26 views

Smoothness in cyclotomic versus complex fields?

Say we have a polynomial in a cyclotomic field; in particular, an n-th cyclotomic field, where n is the order of the polynomial's symmetry group. If we know the polynomial is smooth over this field, ...
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0answers
36 views

How to scale polynomial coefficients for root-finding algorithms?

I've implemented the Jenkins Traub algorithm in c++ (Github repo). While the majority of the solutions work well, it seems that a small portion of the roots are unstable. Here is a link to a ...
0
votes
0answers
44 views

Help with rearranging a large polynomial

I have an expression $$\frac{(-ab)x^2+(a^2+b^2)xy+(-ab)y^2}{2(a-b)^2(0.5(a+b))^2}$$ (a and b are constants) and I need to rearrange it so that it is in the form f(x)*f(y). I am totally lost, and ...
2
votes
4answers
299 views

Determining if function odd or even

This exercise on the Khan Academy requires you to determine whether the following function is odd or even f(x) = $-5x^5 - 2x - 2x^3$ To answer the question, the instructor goes through the following ...
3
votes
1answer
48 views

Solving nonlinear system

I have the following nonlinear system $$\begin{cases} y_1 = \frac{x_1}{\sqrt{x_1^2+x_2^2+x_3^2}} \\ y_2 = \frac{x_2}{\sqrt{x_1^2+x_2^2+x_3^2}} \\ y_3 = \frac{x_3}{\sqrt{x_1^2+x_2^2+x_3^2}} \end{cases} ...
1
vote
1answer
15 views

Integer polynomials and the integration on the unit circle

Let $f (x) $ be a polynomial in $\mathbb {C}[x] $. Then, consider the value $I_f $ defined as $I_f:=\displaystyle\int_0^{\frac {\pi}{2}} |f (e^{ix})|^2 dx$. We can define this on any complex ...
2
votes
5answers
34 views

Is it possible to factor a quadratic equation when $a$, $b$, and $c$ are all equal?

I have the equation $4x^2+4x+4$ to factor. I know that need to start with $$(2x \quad )(2x \quad )$$ to make $4^2$, but I can't seem to factor the rest of the way. What should I do?
3
votes
0answers
41 views

General Fiber in Positive Characteristic

It is well-known that a complex polynomial, considered as a function $f:\mathbb{C}^n\to\mathbb{C}$, is a fiber bundle over a cofinite set of "atypical values" which include the singular values of $f$. ...
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0answers
9 views

Reduce multivariate polynomials by known roots?

Consider three multivariate polynomials $p_1(x,y,z)$, $p_2(x,y,z)$ and $p_3(x,y,z)$ with $x,y,z\in\mathbb{C}$. Imagine that the set of polynomials above is constructed such that they have exactly $6$ ...
6
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0answers
62 views

Is there a polynomial $p$ such that it is bijective and $ p: \mathbb{Q}^n \rightarrow \mathbb{Q}$ for $ n>1$ ??

Let us define a polynomial $p$ defined as follow $$p: \mathbb{Q}^n \rightarrow \mathbb{Q}.$$ I acrossed this question in mind. Is there a polynomial $p$ such that it is bijective and $p: ...
2
votes
1answer
29 views

Degree of Polynomial in Centered Moments of Gamma$(n,1)$

I'm interested in the degree of the polynomial in $n$ of the expression for the $k$-th central moment $$ E((X_n - n)^k) $$ where $X_n$ is a Gamma$(n,1)$ random variable, that is, the sum of $n$ ...
0
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0answers
26 views

A question in matrix polynomial.

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
-1
votes
3answers
150 views

$A$ is a $n \times n$ matrix over $\mathbb{R}$ such that $A^2+A+5I=0$. Find the characteristic polynomial of the matrix $A$. [closed]

$A$ is a $n \times n$ matrix over $\mathbb{R}$ such that $A^2+A+5I=0$. Find the characteristic polynomial of the matrix $A$. it is a question from a test i had yesterday and this is how it was ...
0
votes
1answer
48 views

Is $P(X,Y)=a + aY + (b+cX^2)Y^n \in \mathbb Z [X][Y]$ irreducible?

I am considering the polynomial $a + aY + (b+cX^2)Y^n\in \mathbb Z [X][Y]$, with $n$ even and $a,b,c$ non zero integers. Is this polynomial irreducible or not?
3
votes
1answer
63 views

Factoring bivariate quadratics with real coefficients (for high school students).

I was tutoring a Year 10 student last night (he's learning about quadratics). Unfortunately, we ran into a class of problems that I couldn't explain how to solve (beyond simply guessing and checking), ...