This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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12
votes
2answers
698 views

System of non-linear equations.

I have to find all triplets $(x,y,z)$ that satisfy: $$x^{2012} + y^{2012} + z^{2012} = 3\\x^{2013} + y^{2013} + z^{2013} = 3\\x^{2014} + y^{2014} + z^{2014} = 3$$ I've found the trivial solution $(1,...
1
vote
0answers
71 views

Decomposition of integer polynomials $P(x)$

Let $P(x)$ be an integer polynomial of composite degree $D$. A decomposition is the opposite of a composition. For instance composition of polynomials $A(x),B(x)$ gives $A(B(x)) = C(x)$. The ...
4
votes
1answer
448 views

Deriving Laplace Transform of Laguerre polynomial

I'm given this definition for the Laguerre polynomials: $$L_n(t)=\frac{e^t}{n!}\frac{d^n}{dt^n}\left[t^ne^{-t}\right],~\text{for }n=0,1,2...$$ and I have to show that the Laplace transform is $$\frac{...
3
votes
1answer
61 views

Find integral solutions for $2x^2+y^2=2\times(1007)^2+1$

Find integral solutions to the equation $$2x^2+y^2=2\times(1007)^2+1$$ I tried: I rewrote the equation as $2x^2+y^2=2028099$. I found that $y_{max}=1424$ and $y$ must be odd, so I set $y=1424-(...
1
vote
1answer
52 views

When a given ideal is a radical ideal

I am wondering if there are any canonical methods for checking whether a given ideal is radical. For example, I got stuck on the following example: Let $f=x+2y-z$ and $g=z-2w$ and let $I$ and $J$ be ...
3
votes
2answers
74 views

Derivatives of trig polynomials do not increase degree?

Let $c = \cos x$ and $s = \sin x$, and consider a trigonometric polynomial $p(x)$ in $c$ and $s$. The degree of $p(x)$ is the maximum of $n+m$ in terms $c^n s^m$. Is it the case that repeated ...
1
vote
1answer
94 views

Characterizing maximal ideals in $\mathbb{Z}[x]$

I need to prove this: Let $I\subset\mathbb{Z}$ be the ideal generated by $\{p,f(x)\}$, with $p$ prime in $\mathbb{Z}$. Then $I$ is maximal iff $f(x)$ is irreducible modulo $p$. So I was trying to ...
1
vote
1answer
76 views

hexic polynomial question

I am faced with a polynomial of the form $$ ax^6+bx^3+cx+d=0, $$ where the coefficients are complex. I want to be able to say something about the roots of this polynomial (including finding them!). Is ...
2
votes
4answers
81 views

Finding $p'(0)$ for the polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$

The question goes as follows: Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p'(0)$ is... What I ...
3
votes
1answer
55 views

Uniform convergence of a sequence of polynomial logarithm

Let $P\in \Bbb{C}[X]$ of degree $d\ge 2$. For $n\in \Bbb{N}$ (include $O$). Denote by $P^n$ the $n$-th composition and $g_n: z\mapsto \frac{1}{d^n}\log(\max \{1,\vert P^n\vert\})$. Show that $(g_n)_{n\...
0
votes
2answers
78 views

Solving polynomial to get all coefficients

Given an array of N integers where N can go upto 10^4 and each element can be upto 10^5. Now i need to find the coefficients of polynomial p that is given as : ...
2
votes
1answer
273 views

Division of point-value representation polynomials

In Cormen's "Introduction to algorithms" is exercise: "Explain what is wrong with the “obvious” approach to polynomial division using a point-value representation, i.e., dividing the corresponding y ...
1
vote
1answer
41 views

A “repeated roots allowed” version of the continuity of roots

Let $R_n$ denote the set of all monic real polynomials of degree $n$ all of whose roots are real. Then $R_n$ is a closed subset of the $n+1$-dimensional space ${\mathbb R}_n[X]$. For $P\in R_n$, ...
3
votes
1answer
75 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 ...
3
votes
1answer
67 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 $e$): $a\cdot x^{e}+b\cdot x^{4\cdot e}+c =0$...
3
votes
0answers
48 views

Detect cyclotomic polynomials

I was reading this question: When does a polynomial divide $x^k - 1$ for some $k$? I followed the procedure given by Bill Dubuque in his answer (the "Graeffe" method) for the polynomial $f(x) = x+1$. ...
8
votes
1answer
94 views

If a polynomial has only real zeros then $a_{0}+a_{1}+\cdots+a_{n}\le\frac{(n+1)^n}{\binom{n}{s}(n-s)^{n-s}(s+1)^s}\cdot\max_{k}a_{k}$

Question: For all real polynomials $P(x)=a_{0}+a_{1}x+\cdots+a_{n}x^n$ of degree $n$, with only real zeros,we have $$a_{0}+a_{1}+\cdots+a_{n}\le\dfrac{(n+1)^n}{\binom{n}{s}(n-s)^{n-s}(s+1)^s}\...
1
vote
2answers
1k 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 ...
6
votes
4answers
727 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 &...
0
votes
1answer
63 views

how can I find equation variables?

I have the following equations : $$\begin{cases}K = \frac{B – 3}{20}\\ K = (20S+3)R+S\\ K = 20S^2 + (20N+7)S + N\\ N=S-R \end{cases}$$ - And I have the $B$ values, e.g : 173, 283, 2343, 834343 ...
-1
votes
2answers
50 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 ...
2
votes
1answer
75 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 ...
8
votes
1answer
164 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 ...
1
vote
1answer
221 views

Using resultants to check if multivariate polynomials have a common factor - is my proof correct?

Proposition: Let $f, g \in \mathbb R[x,y,z]$. Then the condition that $f, g$ have a common polynomial factor is an algebraic condition on their coefficients. By algebraic condition, I mean there is a ...
0
votes
0answers
51 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. ...
0
votes
0answers
70 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
3answers
155 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 ...
3
votes
2answers
58 views

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

I need help solving the following equation system: $$ \frac{\partial}{\partial x} = 8xy + 4y^2 + \frac{y}{x^2 + y^2} = 0 $$ $$ \frac{\partial}{\partial y} = 8xy + 4x^2 - \frac{x}{x^2 + y^2} = 0 $$ I'...
0
votes
1answer
74 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 ...
2
votes
2answers
113 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 ...
2
votes
1answer
51 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 ...
1
vote
2answers
129 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 ...
1
vote
3answers
176 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 $(x^2-...
1
vote
1answer
131 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 $\mathbb{R}^...
0
votes
1answer
40 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
votes
1answer
360 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 ...
-1
votes
1answer
92 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 ...
0
votes
1answer
85 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 ...
0
votes
1answer
37 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 ...
8
votes
1answer
191 views

How prove $ \; |f(1)|\le 2004\;$ if $\sqrt {x(1 - x)}\; \Big|f(x)\Big|\le 334$ for $f(x) = Ax^2+ Bx + C $

Let $ \; A,B, C\in {\mathbb R} ,\;$ and $ \; f(x) = Ax^2+ Bx + C$ and $ \sqrt {x(1 - x)} \left|f(x)\right|\le 334,\;\forall x\in [0,1]\;$. How prove $ \; \left|f(1)\right|\le 2004\;$ ?
10
votes
1answer
312 views

Conjecture: Tract version of Gauss--Lucas Theorem for 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 $p',p^{(2)},...
2
votes
1answer
491 views

Factoring homogeneous polynomials in two variables.

Consider a homogeneous polynomial $F(X,Y)\in\mathbb C[X,Y]$. Why can we always write it as: $$F(X,Y)=\prod(a_iX+b_iY)^{r_i}\ ?$$ I can't find a proof of this fact. Many thanks in advance.
2
votes
1answer
63 views

Proving $\sum_{k=0}^n\dfrac{x_k^{n+1}}{\prod_{j\neq k}(x_k-x_j)}=\sum_{k=0}^nx_k$

In Problems from the book by Andreescu, there's the following problem : Let $x_0,\ldots,x_n$ be distinct complex numbers. Prove $\displaystyle \sum_{k=0}^n\dfrac{x_k^{n+1}}{\prod_{j\neq k}(...
1
vote
2answers
58 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 ...
5
votes
3answers
267 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 ...
3
votes
3answers
133 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 ...
4
votes
2answers
280 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, $$f'(x)=[(x-a_2)(x-a_3)\cdots(x-a_n)]+\cdots+[(x-a_1)(x-a_2)\cdots(x-a_{n-1})]$$...
5
votes
4answers
138 views

Solve $(1+z)^8=(1-z)^8$

My guess is to write this as $$\left(\frac{1+z}{1-z}\right)^8=1.$$ We can then find 8 possibilities for $\frac{1+z}{1-z}$, namely $\cos(k\pi/4)+i\sin(k\pi/4)$, $k=1,\ldots,8$. For each $k$ we can then ...
4
votes
1answer
93 views

Motivation of Eisenstein's Irreducibility Criterion

Let $$f(x)=a_nx^n + a_{n-1}x^{n-1}+...+a_0$$ be a polynomial with integer coefficients. If there exist a prime number $p$ such that $$a_{n-1} \equiv a_{n-1} \equiv...\equiv a_0 \equiv 0 \pmod p$$ but $...
0
votes
1answer
185 views

Why use regularization to reduce over-fitting

I'm having trouble understanding why should we use regularization for over-fitting when we can simply reduce the number of order to our polynomial function? Is it because it saves us time from having ...