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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3
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1answer
65 views

Analytical solution of a polynomial $a\cdot x^{e}+b\cdot x^{4\cdot e}+c =0$

Is it possible to get an analytical solution of the equation $a\cdot x^{2\cdot e}+b\cdot x^{e+1}+c =0$ Which can be also written as (due to the value of $e$): $a\cdot x^{e}+b\cdot x^{4\cdot e}+c ...
0
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2answers
35 views

How does the cross multiplication of Quadratic Equation work?

How does the cross multiplication of Quadratic Equation works? If: $$f_1\left(x\right)=a_1x^2+b_1x+c_1=0$$ and: $$f_2\left(x\right)=a_2x^2+b_2x+c_2=0$$ have a common root, let's say, $\alpha$, then ...
0
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1answer
41 views

Find polynomials $f (x)$, $g(x)$, and $h(x)$

In an elementary Algebra book (101 problems in Algebra) there was a question I solved but when I looked at the solutions I didn't get it. it says find Polynomials $f(x)$, $g(x)$, $h(x)$ such that for ...
2
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1answer
66 views

Method to simplify this long expression

How can I simplify this long expression: $-a^3(d-b)(d-c)(c-b)+b^3(d-a)(d-c)(c-a)-c^3(d-a)(d-b)(b-a)+d^3(c-a)(b-a)(c-b)$ I know that it is equal to $(d-a)(d-b)(d-c)(c-a)(c-b)(b-a)$ but i have no idea ...
6
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4answers
374 views

Determinant of a matrix with $t$ in all off-diagonal entries.

It seems from playing around with small values of $n$ that $$ \det \left( \begin{array}{ccccc} -1 & t & t & \dots & t\\ t & -1 & t & \dots & t\\ t & t & -1 ...
3
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1answer
329 views

Checking if a System of Polynomial Equations is Consistent

I'm trying to determine whether any solutions exist to a system of $(n+1)$ polynomial equations in $n$ unknowns. For example: $$ \begin{align*} xy&=-2\\ x^2-1&=0\\ y^3-3y^2+2y&=0 ...
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0answers
11 views

Simultaneous bi-variant polynomial equations

I'm having trouble with solving out a pair of (what I'm thinking of as) bivariate polynomial equations. I'm just looking to express $x$ and $y$ as functions of $A, B, C, D$, though my fruitless search ...
2
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1answer
107 views

Inverse of a matrix is expressible as a polynomial?

Let $A$ be an $n \times n$ matrix. Prove that if A is invertible, then there exists a polynomial $p$, such that $A^{-1}=p(A)$ Thus far: Let $W$ denote the $k$ dimensional A-cyclic subspace spanned ...
5
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0answers
143 views

How control small perturbations keeping a zero of a polynomial?

Let $P : \mathbb{R}^n \to \mathbb{R}^m$ by a polynomial function. Let $x_0 \in \mathbb{R}^n$ be a zero of $P$ such that the graph of $P$ crosses $\mathbb{R}^n \times \{ \vec{0}\}$ around ...
4
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1answer
133 views

Fast way to find the smallest root $\mod M$ of a polynomial

Suppose you're given a polynomial of degree $d$ with integer coefficients: $$ P(x) = \sum_{i=0}^{d}{a_i x^i} $$ Is there a fast way to find the smallest root modulo $M$, where $M$ is some composite ...
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0answers
81 views

Cardano, Descartes, Linear Equations and polynomials of degree greater than 3 [closed]

How would you describe the physical significance of algebraic equations of degree $> 3$, and generally of polynomials of degree $> 3$ ? One line of thought is that $x$ is length, $x^2$ is area ...
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2answers
37 views

How to get A,B and C given XYZ?

How do I get $a$, $b$, and $c$? Given $$X=\frac{a+\frac{1}2b}{a+b+c}$$ $$Y=\frac{b(\frac{\sqrt3}{2})}{a+b+c}$$ $$Z=\frac{a+b+c}{3}$$ in other words How do i get $a$, $b$, and $c$ on the left ...
3
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3answers
2k views

Polynomials, derivatives and repeated roots

I want to describe the polynomials with integer coefficients and the property that $f'(x) \mid f(x)$ (the derivative divides the polynomial). So I know that $f(x)$ divides $g(x)$ if all of ...
7
votes
1answer
146 views

Do perfect polynomials of degree $4$ exist?

I asked this question already, but I cannot find it anymore. If it is a duplicate, I will delete it. Is there a polynomial $$p(x)=x^4+ax^3+bx^2+cx+d$$ such that p and all the derivates upto the ...
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0answers
11 views

unknown function: calculation of coefficients in series expansion up to a given degree

I am trying to solve an functional equation of unknown $h\mapsto h(x)$ ($x\in\mathbb{R}$, in the neighbourhood of $0$): $$\mathcal{F}(h)=\mathcal{G}(h) \qquad (*)$$ (assume $\mathcal{F}$ and ...
0
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0answers
29 views

Eisenstein's Criterion — why $p$ has to be prime?

To prove the validity of the criterion, suppose $Q$ satisfies the criterion for the prime number $p$, but that it is nevertheless reducible in $Q[x]$, from which we wish to obtain a contradiction. ...
4
votes
1answer
281 views

Rank of a rectangular Vandermonde Matrix to which weighted columns are added

A Vandermonde matrix: $\left(\begin{array}{ccc} 1 & \alpha_{0} & \dots & \alpha_{0}^{n} \\ 1 & \alpha_{1} & \dots & \alpha_{1}^{n} \\ \vdots & \vdots & \ddots & ...
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0answers
38 views

Why do nth roots (radicals) have closed forms whilst other polynomial roots do not?

The nth root function - $\sqrt[n]{a_{0}}$ - may be seen as an arithmetic operation (the inverse of the $pow(x, n)$ function) but it can also be interpreted as computing the roots of a specific class ...
5
votes
1answer
69 views

A polynomial and its degree

What is the degree and the value of the leading co-efficient of the polynomial $$\prod_{j=0}^{\lfloor{n/2}\rfloor}(x+2j+1)^{\binom{n}{2j+1}}-\prod_{j=0}^{\lfloor{n/2}\rfloor}(x+2j)^{\binom{n}{2j}} ...
6
votes
3answers
115 views

Is $x^x$ a polynomial, an exponential or both?

If $c$ is a constant, and $x$ is a variable, we'd say that $f(x) = x^c$ is a polynomial function of order $c$. Conversely, the function $f(x) = c^x$ would be called an exponential function. Is there ...
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2answers
60 views

Linear Algebra - Given the Jordan form of $A \in Mat_7(\mathbb F)$, find Jordan form of $A^2+A+I_7$

Given that the jordan form of the matrix $A \in Mat_7(\mathbb F)$ is: $\begin{pmatrix} J_2(1) &\cdots &0\\0& \cdots J_3(1) \cdots &0\\0&0& \cdots J_2(2)\end{pmatrix}$ Find ...
3
votes
2answers
135 views

Closed form of a sum of binomial coefficients?

I have the following function: $T_n(d)=\sum\limits_{k=\frac{n-d}{2}}^{\lceil \frac{n}{2} \rceil}{k\choose \frac{n-d}{2}}$ ${n \choose 2k}$, where $n,d\in \mathbb{N}^0$, and $n,d$ have the same ...
7
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2answers
301 views

Perron polynomial irreducibility criterion

Facts before question: $\textbf{Fact 1:}$ Let $F(X) = X^n + a_{n-1}X^{n-1} + \cdots + a_1X+a_0\in \mathbb{Z}[X]$, with $a_0\neq 0$. If $|a_{n-1}|>1+|a_{n-2}| + \cdots +|a_1| + |a_0|$, then $F$ ...
0
votes
0answers
7 views

Solutions to a polynomial equation in a PAC field not lying in a subfield

Suppose $f(x,y)$ is an absolutely irreducible polynomial over a PAC (pseudo algebraically closed) field $K$ such that $x,y$ actually appear in $f$. Let $L$ be a proper subfield of $K$. Are we ...
4
votes
0answers
92 views

Level curves of a polynomial and the zeros of its higher derivatives.

The Gauss--Lucas Theorem states that all zeros of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros of ...
2
votes
1answer
43 views

Facts about quotient rings - example

I have three quotient rings: $R_1 = \frac{\mathbb{Q}[x]}{(x^2 -1)}$ $R_2 = \frac{\mathbb{Q}[x]}{(x^2 +1)}$ $R_3 = \frac{\mathbb{Q}[x]}{((x -1)^2)}$ I am trying to decide whether these are integral ...
3
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2answers
39 views

Solving $4y^4 - 4x^4 + x + y = 0$ (equation system of partial derivates)

I need help solving the following equation system: $$ \frac{\partial}{\partial x} = 8xy + 4y^2 + \frac{y}{x^2 + y^2} = 0 $$ $$ \frac{\partial}{\partial y} = 8xy + 4x^2 - \frac{x}{x^2 + y^2} = 0 $$ ...
0
votes
1answer
44 views

How to solve a nonlinear system of three equations involving rational functions?

How do I get $a$, $b$, and $c$? Given $$X=\frac{a+\frac{1}2b}{a+b+c}$$ $$Y=\frac{b(\frac{\sqrt3}{2})}{a+b+c}$$ $$Z=\frac{76a+150b+29c}{255}$$ in other words How do i get $a$, $b$, and $c$ on the ...
0
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1answer
40 views

Remainder theorem for a real polynomial [closed]

A certain polynomial $p(x)\in\mathbb R[x]$, when divided by $x-a$, $x-b$, $x-c$ gives remainders $a$, $b$, $c$, respectively. How can I find the remainder when $p(x)$ is divided by $(x-a)(x-b)(x-c)$ ...
0
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1answer
60 views

How do I reverse this formula

How do I get $a$, $b$, and $c$ given $$X=\frac{a+\frac{1}2b}{a+b+c}$$ $$Y=\frac{b(\frac{\sqrt3}{2})}{a+b+c}$$ $$Z=\frac{76a+150b+29c}{255}$$ in other words How do i get $a$, $b$, and $c$ on the ...
2
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2answers
68 views

“Conic sections” that are really just two straight lines

My teacher was teaching co-ordinate geometry and today he said that the following equation will always represent a conic section:$$ax^2+by^2+2hxy+2gx+2fy+c=0$$ Then he said that if the determinant of ...
0
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1answer
65 views

Determine the nature of $f(x)$

Consider a polynomial $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients. Determine and prove the nature of $f(x)$. Can someone ...
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3answers
65 views

Find polynomial whose root is sum of roots of other polynomials

We have two numbers $\alpha$ and $\beta$. We know that $\alpha$ is root of polynomial $P_n(x)$ of degree $n$ and $\beta$ is root of polynomial $Q_m(x)$ of degree $m$. How do you find polynomial $R_{n ...
2
votes
1answer
32 views

Strict local extremum without $f'$ “changing signs”

Let $f:\mathbb{R}\to \mathbb{R}$. Is it possible that $f$ has the following properties: $f$ is differentiable in a neighborhood of $a\in \mathbb{R}$ $a$ is a strict local minimum There is no ...
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2answers
28 views

Do polynomials $ P(t)$ of an odd degree have at least one real root belong to $(t-a)Q(t)$?

This is a continuation of a question where ker(T) = (t-a)Q(t) = P(t). Show that {P(t) ∈ R[t] | deg(P(t)) = 3} ⊂ $∪_{a∈R}$ker(T). So the mark scheme says that all polynomials in R[t] of an odd ...
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2answers
130 views

Algorithms for factoring multivariate polynomials

I am wondering if there are any algorithms to factor polynomials in multiple variables, when you know that the factors are other polynomials with rational or integer coefficients. I know you have the ...
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1answer
31 views

A question on the standard basis for polynomials

I'm trying to self-study Linear Algebra from Linear Algebra Done Wrong, but the book hasn't explained everything properly so my question might be extremely easy, apologize in advance: For ...
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3answers
47 views

The number of ideals in the quotient ring $\mathbb R[x]/\langle x^2-3x+2 \rangle$ [duplicate]

Finding the number of ideals in the quotient ring $\mathbb R[x]/\langle x^2-3x+2 \rangle$. Attempt: $R[x]/\langle x^2-3x+2 \rangle = \{f(x)+\langle x^2-3x+2 \rangle~~|~~f(x) \in R[x]\}$. Since ...
3
votes
2answers
555 views

How to demonstrate that there is no all-prime generating polynomial with rational cofficents?

It seems like there is no polynomial with finite variables known, which could generate all prime numbers, by integer assignments. Is there a proof that such polynomial can not exist and does anyone ...
0
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0answers
10 views

For what values of λ is this family free (independent), spanning and a basis of R[t]≤3

The family of polynomials $F$ = {${(λ^2 − 1)t^3 + t^2, λt^3 + t − λ, (1 − λ)t^3 + t + 1, λ}$} in $R[t]_{≤3}$ I set their sum to 0 to find the values for it to be independent. $a((λ^2 − 1)t^3 + t^2) ...
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1answer
39 views

Specify the values of $p$ and $p'$ for a polynomial

Problem 10-26 from Spivak's Calculus, 4th edition: Let $a_1, \dotsc, a_n$ and $b_1, \dotsc, b_n$ be given numbers. If $x_1, \dotsc, x_n$ are distinct numbers, prove that there is a polynomial ...
0
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1answer
18 views

Marking the roots of a quadratic function in Scilab

I have 2D plotted a simple quadratic function in Scilab and now have to mark the roots with an X. Is there any simple way of doing that? I have written a function that calculates the roots and ...
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2answers
40 views

local inverse of polynomial

Is there a possibility to invert a polynomial locally? I've got the following problem, concerning control theory: Imagine a ideal amplifier with a feedback loop: Let firstly A be not dependent on ...
0
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1answer
20 views

Plotting three variables on an XY plane, involves distance formula.

I have 3 dynamic constants with values of 0 to 1. Lets label them A,B and C. I want to be able to plot them on a 2 dimensional cartesian plane. so given all three constants I will be able to find the ...
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2answers
262 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
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2answers
218 views

Is there a rule to the terms of a falling factorial?

$\require{cancel}$I discovered that $n!=\xcancel{(n)_{n-1}}n^{\underline{n-1}}=n(n-1)(n-2)\cdots(3)(2)$. I have expanded a few examples: $$2!=\xcancel{(2)_1}2^{\underline{1}}=2\\ ...
0
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2answers
95 views

How to show that $f(x)$ is never negative

$f(x)=x^4-2x^3-2x^2+a$ When $a=8$, show that $f(x)$ is never negative. $x^2-4x+4$ is a factor of $f(x)$. How do I work out this question? I have tried using the factor theorem but I honestly ...
0
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2answers
98 views

How to solve the cubic equation $ x^3+3x -2 = 0$ without using matrices?

I am trying to solve $ x^3+3x -2 = 0$ Using the remainder theroem but none of the factors of the constant make the equation equal to $0$. Is there any way I can get the answers without using matrices? ...
1
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2answers
92 views

Determine the polynomials $p(x)$ satisfying $x\cdot p(x-1) = (x-26)\cdot p(x)$

Determine the polynomials $p(x)$ satisfying $x\cdot p(x-1) = (x-26)\cdot p(x)$. My Solution: Put $x=0$, we get $p(0) = 0$, Similarly put $x=26,$ we get $p(26) = 0$. That means $x=0,26$ are two roots ...
3
votes
2answers
154 views

Discriminant of $x^n-1$

Question is to find discriminant of polynomial $x^n-1$ I consider $f(x)=x^n-1=(x-a_1)(x-a_2)(x-a_3)\cdots(x-a_n)$ Now, ...