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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Are the conditions ${c=ae+(b-e), d=e(b-e)}$ and $ad=(b-e)[c-(b-e)]$ equivalent?

Respected All. Please help me on the following. Suppose that $x^5+ax^4+bx^3+cx^2+dx+e\in \mathbb R[x]$ can be written as $$(x^3+a_1x^2+b_1x+1)(x^2+b_2)$$ then we must have \begin{align*} ...
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1answer
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Finding the matrix representation of a linear transformation $ T: P_{3} \to \text{M}_{2 \times 2} $.

Define a function $ T: P_{3} \to \text{M}_{2 \times 2} $ by $$ T \! \left( a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} \right) = \begin{pmatrix} a_{3} & a_{0} \\ a_{2} & a_{1} \end{pmatrix}. ...
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Ideals and multivariable polynomials

Prompt: Let f[x,y] denote the domain of all the polynomials \sum (a_{ij}x^{i}y^{j})\ in two letters x and y, with coefficients in F, where F is a field. Let J be the ideal of f[x,y] which contains all ...
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2answers
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Polynomials of degree less than $n$ that agree at $n$ values

Suppose $\deg(a(x))$ and $\deg(b(x))$ are both less than $n$. If $a(c) = b(c)$ for $n$ values of $c$, prove that $a(x) = b(x)$. This seems simple, since if the $a(c) = b(c)$ for $n$ values of ...
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2answers
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Lowest degree polynomial satisfying $\ P(n)=1/n$ for first n natural numbers

So say I wanted to find the lowest degree polynomial satisfying$\ P(1)=1, P(2)=1/2, P(3)=1/3$. Is there some sort of formula where I can put in$\ n$ and there will be a polynomial with coefficients in ...
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1answer
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Find the roots of the polynomial in $Z_5$

The polynomial is: $2x^{219} + 3x^{74} + 2x^{57} + 3x^{44}$. Find the zeros. Now my first step, which I believe shall be correct is to reduce the exponents of the polynomial in mod 5. Thus: 219 ...
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1answer
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Vanishing set of irreducible polynomials

Question: Find irreducible $f,g \in \mathbb{R}[x,y]$ such that $V(f) = V(g) \neq 0$ with the added requirement $f \neq \lambda g$ for $\lambda \in \mathbb{R} - \{0\}$. Attempt: I think $f(x,y) = x^2 ...
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1answer
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Solve polynomial equation in $\mathbb{C}[x]$

Find the polynomials $f,g \in \mathbb{C}[x]$ with complex coefficients such that: $$f(f(x))-g(g(x))=1+i,\\f(g(x))-g(f(x))=1-i$$ for all $x\in\mathbb{C}$. I think I have this problem almost ...
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30 views

Assume that d,u,q are all positive integers. Also, d-u is divisible by q, u-q is divisible by d and q-d is divisible by u.

Assume that d,u,q are all positive integers. Also, d-u is divisible by q, u-q is divisible by d and q-d is divisible by u. What is the solution for $d,u,q$? Am I right to assume that the solution is ...
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How should I create a single score with two values as input?

I have two series of values, a and b as inputs and I want to create a score, c, which reflects both of them equally. The distribution of a and b are below In both cases, the x-axis is just an ...
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32 views

Observation about Polynomials Addition

For two polynomials $f_1=(x-a)(x-b), f_2=(x-a)(x-e)$, if we add them together: $f_3=f_1+f_2$, $f_3$ only has an integer root that is $a$. I observed that it'd possible to make $f_3$ have more than one ...
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1answer
40 views

Sums of two irreducible polynomials over $\mathbb{Z}$

Please help me to prove that any polynomial with integer coefficients can be represented as a sum of two irreducible polynomials over the ring $\mathbb{Z}$.
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Bound on roots of a monic polynomial

Let $p(x)$ be a real monic polynomial of degree $n$. The following information is given about $p(x)=0$: all roots are real there is no zero root exactly $k$ roots are positive and $n-k$ roots are ...
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Prove $p(x)>0$ for $x>b$

This is a question from a past paper which I have no solution to. Let $p(x)=x^n + a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}, n\geq 1$ be a polynomial of dgree n and let $b=|a_{1}|+\cdots ...
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1answer
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quadratic form polynomial divisibility vs. matrix pointwise multiplication. [on hold]

Given matrix $V',W',Y'$ is of $d\times m (d\le m)$ ; column vector $c$ is of size $m$; $r_i, i=1,...,d$ are distinct; and each row of the matrix A is $A_i=(r_i^0 ... r_i^{d-1})$. So, A is of $d\times ...
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2answers
32 views

Checking irreducibility of a polynomial over a finite field

A part of a coding theory course I am doing includes some questions on irreducible polynomials. I have a question with solution but am worried I have interpreted it incorrectly. So for $\mathbb F_5$ ...
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0answers
53 views

Find all positive values for j,k,l such that j, k, l are positive integers and (j-k)|l, (k-l)|j, (l-j)|k.

Find all possible values of $j,k,l$ such that $j, k, l$ are positive integers and $(j-k)|l, (k-l)|j, (l-j)|k$. As I understand that using divisibility properties, it is possible to come to some ...
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2answers
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Find a second root of $x^3+px+q$ given the first root

This is a problem from Artin where given one root $a$, you have to find an equation for a second root in terms of $a$, $p$, $q$, and the square root of the discriminant $\delta$. Here's what I have ...
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Is the Möbius inversion applicable in the case of number functions with values in $Q(x)$

I am looking for the cause of an erroneous calculation I did the details I cant present here. I guess a "Möbius inversion" I apply might be the cause. Normally the Möbius inversion is valid for ...
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1answer
58 views

Solving polynomial equation using Fermat little theorem

I am a bit confused on notation. I can't find a reference in notation in my textbook as to what this means. Here it goes: Let p = 13. Compute $\phi$$_{11}$$(3x^{233} + 4x^6 + 2x^{37} + 3)$ This ...
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Every polynomial has a root

Let $A$ be a commutatif ring, and $f\in A[T]$ une polynome. Then in the $A$-algebre $B=A[T]/(f)$ the polynomial $f$ has a root, namely $T \mod (f)$, because $f(T)\mod (f)=f(T)\mod (f)=0$. Do you ...
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Is there a formula which would let me know how many irreducible polynomials there are to the power n, in $z_n$? [duplicate]

I found that $x^2+x+1$ is the only polynomial to the power 2 that is irreducible in $z_2$. Moreover I found that $x^3+x+1$ and $x^3+x^2+1$ are the only polynomials to the power 3 that are ...
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4answers
46 views

The complex roots of a biquadratic polynom

In my recent post I have a problem with the following function: $x^4-4x^2+16$, and what I need is to find the complex roots. Here is my answer: First step, I make the substitution $x^2=y$ which ...
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0answers
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Faber polynomials in Matlab [on hold]

I want to compute some faber polynomials associated to an ellipse centered at a point \sigma (in the complex plane) in Matlab. Say the ellipse has minor axis a and major axis b. If someone know how to ...
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Is the multiplication modulo $p$ for polynomials well-defined?

Is the multiplication modulo $p$ for polynomials well-defined ? I mean let $g,h\in\mathbb Z[x]$ and let $\bar g$ be the polynomial obtained from $g$ by reducing all the coefficients of $g$ modulo ...
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801 views

Is Vieta the only way out?

Let $a,b,c$ are the three roots of the equation $x^3-x-1=0$. Then find the equation whose roots are $\frac{1+a}{1-a}$,$\frac{1+b}{1-b}$,$\frac{1+c}{1-c}$. The only solution I could think of is by ...
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2answers
681 views

Why is the zero polynomial not assigned a degree?

Yesterday, I read in my textbook, We assign degree to every polynomial and even a non-zero constant is assigned a degree $0$ but $0$ itself is not assigned a degree. Why is that? Why we don't ...
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1answer
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Location of the roots of $f'$ (Laguerre's theorem)

Let $f \in \mathbb{R}[X]$ be a polynomial of degree $n$ having $n$ distinct roots $a_1,...,a_n$. Let $b_1<...<b_{n-1}$ be the roots of its derivative $f'$ (note that $b_i \in ]a_{i}, a_{i+1}[$ ...
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If $\alpha$ and $\beta$ are the zeros of the polynomial $p(s)=3s^2-6s+4$, find the value of

If $\alpha$ and $\beta$ are the zeros of the polynomial $p(s)=3s^2-6s+4$, find the value of $\frac{\alpha}{\beta}$+$\frac{\beta}{\alpha}$+2$(\frac{1}{\alpha}$+$\frac{1}{\beta})$+3$\alpha\beta$ By ...
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1answer
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existence of multiplicity of roots [on hold]

Im confuse..I read in an article that in dealing with polynomials, a quadratic equation can have either 2 real roots, 1 equal real root or 2 complex roots...but in dealing with random polynomials only ...
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1answer
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Why is $z^4-1-i=0$ a polynomial equation which does not have real coefficients?(complex-number)

Why is $z^4-1-i=0$ a polynomial equation which does not have real coefficients? Its coefficient is $1$ and $1$ is a real number, isn't it?
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Binary division using polynomial

I want to do a division of two binaries and take the rest (mod). But I want to do this using polynomials, let's take the example: binary dividend: 010001100101000000000000 binary divisor: 100000111 ...
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1answer
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Division binary using polynomials

I want to do a division of two binaries and take the rest (mod). But I want to do this using polynomials, let's take the example: binary dividend: 010001100101000000000000 binary divisor: 100000111 ...
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0answers
48 views

Corollary of Gauss's Lemma (polynomials)

I am trying to prove the following result. I have outlined my attempt at a proof but I get stuck. Any help would be welcome! Theorem: Let $R$ be a UFD and let $K$ be its field of fractions. ...
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Express homogeneous polynom on unit sphere by higher-degree polynom

I have come across the following statement in 10.1007/BF02391776 (Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds, p. 159, middle): ...
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Closed-form for one of the solutions of a specific polynomial equation of degree five of higher with integer coefficients [duplicate]

Because of Abel–Ruffini theorem we know that there is no solution in radicals to polynomial equations of degree five or higher with arbitrary coefficients. For example we know that there is no ...
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34 views

Show: Real roots of a polynomial

I´ve got some problems with this task I found in a german script. Be $\ N \in \mathbb{N}$. Define matrix $\ \Pi := (\pi)_{0\leq i,j \leq N}$ with \begin{equation} \pi_{i,j} := \binom{N}{j} p_i^j ...
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Factor a polynomial

How do I factor this polynomial: $$x^4-x^2+1$$ The solution is: $$(x^2-x\sqrt{3}+1)(x^2+x\sqrt{3}+1)$$ Can you please explain what method is used there and what methods can I use generally for 4th or ...
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Compute the largest root of $x^4-x^3-5x^2+2x+6$

I want to calculate the largest root of $p(x)=x^4-x^3-5x^2+2x+6$. I note that $p(2) = -6$ and $p(3)=21$. So we must have a zero between two and three. Then I can go on calculating $p(\tfrac52)$ and ...
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Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible

Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible. I am trying to prove it with Eisenstein's criterion but without success: for p=2, it devide -4 and the constant coefficient 16, don't devide the ...
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Composition of polynomials over finite fields

Consider the set of polynomials of degree at most $n$ over a finite field $k_q$ with $q$ elements where $q$ is prime: $$ P_{n,q} = \left\{ x + c_2 x^2 + \cdots + c_n x^n:\ c_i \in k_q \right\}. $$ It ...
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1answer
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X - axis of a linearized polynomial.

The other day in my Physics class we had some sample data that we wanted to linearize. The graph resembled a root curve. So to linearize it, we took the square root of all the x data and replotted ...
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Uniform convergence of polynomials (including first and second derivatives)

I am searching for a proof of the following statement: Given a twice continuously differentiable (real-valued) function on $\mathbb{R}^n$ and a compact set $K$, one can find a sequence of polynomials ...
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1answer
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Find the number of divisors of $f'(1)$

The question is that: Let $f(x) = x^{25} + 2x^{24} + 3x^{23} + 4x^{22} + \cdots + 25x$. Find the number of positive divisors of $f'(1)$. How to find this number easily? Is there only one way: ...
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2answers
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Extreme point of quadratic equation

For the below question read here: Write a function quadratic that returns the interval of all values $f(t)$ such that $t$ is in the argument interval $x$ and $f(t)$ is a quadratic function: ...
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1answer
30 views

Exist another method to solve the problem?

We have $x_1,\:x_2,\:x_3\:\in \:\mathbb{C},\:\:f=x^3+x^2+mx+m,\:m\in \mathbb{R}$. We need to find $m\in\mathbb{R}$ such that $|x_1|=|x_2|=|x_3|$. Here is what I tried: $f=x^3+x^2+mx+m=(x^2+m)(x+1)$, ...
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335 views

Proving a polynomial has a solution in the interval (0,1)

I have no idea how to start this problem. I am assuming that the Mean Value Theorem is needed in the proof but I am not exactly sure how to apply it to the given polynomial. Any hints/help would be ...
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1answer
71 views

Inverting a map from a finite 3D grid to 1D

I need help with this mathematics question. I have made a program on the computer that flattens a 3D array into a 1D array. A 3D array needs an x, y and z but by using this formula (max x * max y * ...
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1answer
22 views

Reducible Polynomials in finite fields

I am stuck on the following question. Verify that $x^5 + x + 1$ is reducible in $Z_2[x]$ and find its factors. Help would be much appreciated whether it is the answers with how you did it or just the ...
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2answers
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Working out a polynomial from it's solutions when set equal to zero

If I have a polynomial of degree $n$ with leading coefficient $1$, that when set equal to zero has as its only solution $x=0$, how do I prove that this polynomial can only be $x^n$?