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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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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1answer
67 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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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 ...
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21 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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93 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 ...
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1answer
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Factoring homogeneous polynomials in two variables.

Consider a homogeneous polynomial $F(X,Y)\in\mathbb C[X,Y]$, why we can always write it as: $$F(X,Y)=\prod(a_iX+b_iY)^{r_i}\quad?$$ I can't find a proof of this fact. Many thanks in advance.
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1answer
47 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 ...
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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 ...
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3answers
154 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 ...
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66 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 ...
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157 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, ...
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1answer
21 views

Factor the equation either by pairs method or any other [closed]

I tried to split by pairs but i got nowhere. $$x^2 - 4y^2 - 4x + 4$$
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1answer
47 views

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 ...
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1answer
56 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 ...
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3answers
39 views

where am I going wrong with solving this equation?

solve $z^2=2e^{5{\pi}i/6}$. Well, clearly $z={\sqrt{2}}e^{5{\pi}i/12}$ is a root so its' conjugate $z={\sqrt{2}}e^{-5{\pi}i/12}$ is the other root. But I can also argue ...
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1answer
71 views

A better definition of Polynomial

Usually, we define a polynomial as $a_n x^n + \cdots + a_1 x + a_0$ where $x$ is called indeterminate. Would it be better to define it as $a_n x^n + \cdots + a_1 x + a_0 x^0$ where $x^0$ means ...
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Find the sum of real roots of a biquadratic equation

Given the following biquadratic equation: $$x^4-3x^3-2x^2-3x+1=0$$ Find the sum of its real roots. Let $$f(x)=x^4-3x^3-2x^2-3x+1$$ By observing the behaviour of $f^{'}(x)$, I was able to deduce that ...
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0answers
11 views

Reading on Laurent Polynomials

I'm interested in reading about Laurent Polynomials. Does anyone know a good resource/book that I can read about Laurent polynomials? Thanks.
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61 views

Show that $\sqrt{2}$ is irrational using integer root theorem

Show that $\sqrt{2}$ is irrational using integer root theorem. Let $P(x)=x^2-2$. Since $\sqrt{2}$ is a root of this polynomial, had it been a rational (suppose $\sqrt{2}=\frac{p}{q}$) no, by ...
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2answers
26 views

How to Calculate the Sum of Coefficients in a Polynomial with known Integer Roots

I have this problem: given $N$, $1 \leq N\leq 100$ integers which are roots from a polynomial, calculate the sum of coefficients from that polynomial for example: given $3$ integers $2$, $2$ and ...
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Solve in $\mathbb{R}$: $(x^2-3x-2)^2-3(x^2-3x-2)-2-x=0$

Solve in $\mathbb{R}$: $(x^2-3x-2)^2-3(x^2-3x-2)-2-x=0$ I'm supposed to solve this equation. It's from a math contest so solving it by hand would be preferable (no quartic formulas). I thought ...
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Generator polynomial of general cyclic codes

I 'm new to finite field and ring ideals. I notice that any cyclic codes can be generated from a generator polynomial $g(x)=\prod_{i}M_i(x)$, where $M_i(x)$ are some minmal polynomials over some ...
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2answers
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Multiplicity of roots in finite fields of order prime. [closed]

I am having trouble with completing this question from last years exam (part a and d) Let p be a prime, and $f = x^5-1 = (X-1)(X^4+X^3+X^2+X+1) \in \mathbb{F}_p[X]$ Show: (a) if $p\neq 5$ then every ...
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78 views

Factoring a problem. What is the other factor? [closed]

One of the factors of the polynomial $x^3-5x^2$ is $x+3$. What is the other factor
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3answers
65 views

generalized way of finding pair solutions of an equation

I want to find out pair solutions of this equation: $$x^{2}-79y^{2}=1$$ This is a hyperbola equation. I sketched its graph, but that didn't help me. I think the square from (form?) of $x$ and $y$ is ...
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2answers
81 views

Connection with golden ratio?

Consider the following problem: Let $p\in\mathbb{Z}[x]$ be a polynomial with integer coefficient. Suppose that the leading coefficient is 1, all roots are real and in $(0, 3)$. Find all ...
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2answers
30 views

Bounded number of multiple roots in a family of polynomials

Let $F$ be a field and let $f(x)$ be a fixed nonconstant polynomial. Look at the family of polynomials $x^n - f(x)$, where $n$ varies. It is reasonable to assume that these have $n - C$ distinct roots ...
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38 views

Creating a polynomial function with no x-intercept

I have an understanding of polynomials and how to create a function based on the leading coefficient, degrees, x-intercepts, etc. My question is how do i go about creating a polynomial function that ...
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37 views

Solution of $(1-x)^p= x$ in $(0,1/n)$

Let $n,p$ be positive integers. The equation $$ (1-x)^p = x $$ has a unique solution $x_p$ in the interval $(0,1)$. This follows by the monotonicity properties of $(1-x)^p$ and $x$. My question is: ...
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3answers
92 views

Evaluate $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at $t=1$

I need to find a "nice" formula for the evaluation of $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at t=1, where $d_j \in \mathbb{N}$. I have already proved ...
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0answers
60 views

GCD of high order polynomials(modulo large prime)

I want to solve the following question: Consider a polynomial $f(x)=a_0+a_1*x^{e_1}+a_2*x^{e_2}+\cdots+x^{e_m}\in F_p[x]$ where $p$ is a prime such that $\log(p)\sim m$ and $e_m\sim 2^m$, I want to ...
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2answers
39 views

Degree of Field Extension over $\mathbb{F}_2$

While studying for quals, this question came up: Let $K = \mathbb{F}_2$, and $\alpha^{17} = 1$ where $\alpha \in \overline{K}$. If $\alpha \neq 1$ then $K(\alpha)/K$ has degree $8$. This ...
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1answer
43 views

How to obtain the number of real valued zeroes of a polynomial?

While I know there's no analytical formula for the roots of a general polynomial of degree five and higher, I wonder whether there is at least something like a parabola's discriminant to determine how ...
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192 views

Families of Polynomials Irreducible in $\mathbb{Z}$ but reducible in $\mathbb{Z}/p\mathbb{Z}$ for all primes $p$.

I am wondering if there exist classification of polynomials that are irreducible in $ \mathbb{Z}$ but reducible $\pmod p$ for all primes $p$. I am aware that $\Phi_n$ has this property if ...
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2answers
57 views

For $5$ distinct integers $a_i$, $1\le i\le5$, $f(a_i)=2$. Find an integer b (if it exists) such that f(b) = $9$.

Here's an interesting question I came across.The person who gave it to me told me that it should not take more than $3$ minutes to solve this question. But I could not find any definite solution :( ...
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3answers
71 views

Does substitute $\lambda$ with matrix $A$ in a polynomial conflict with the Axiom of Substitution?

This seems to be an elementary question, gonna ask it anyway. Suppose that $A$ is a square matrix, and that $p(x)$ is its characteristic polynomial, we know that (1) $p(x) = \det(xE - A)$ We also ...
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Interpolation point selection for Rational Polynomial Interpolation

people, 1st time on math.stackexchange so aloha to all!! The Question: I have a certain data set and I am using Thiele's rational Polynomial Interpolation to interpolate some data but the curve will ...
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2answers
51 views

Integer roots to cubic equation

If I have a cubic equation $x^3 + ax^2 + bx + c = 0$, what constraints exist on $a,b,c$ when we have three integer solutions? How do I choose $a,b,c$ to force integer solutions?
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2answers
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Trigonometric identity on $\cos \pi/7$ [duplicate]

I found this in a book I used to train myself for the Math Olympics a bunch of years ago: Prove that $$\cos\frac{\pi}{7}-\cos\frac{2\pi}{7}+\cos\frac{3\pi}{7}=\frac{1}{2} $$ I couldn't solve it ...
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Eigenvalue formula for 4x4 symmetric matrix

Is there a formula/algorithm that is accurate to used in finite precision arithmetic (aka numerical stable ) for small symmetric matrix of size 4x4. Additionally I'm looking if it require similar ...
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6answers
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How to find the roots of $-x^3+3x^2-7x+5 = 0$?

I would like to understand how to go about solving something like this, not just get the solution but some kind of methodology (that hopefully makes as much intuitive sense as possible); I honestly ...
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1answer
52 views

Nice polynomial reducibility: $x^n+4$

Problem: Find all $n\in \mathbb{N}$ such that $f(x)=x^n+4$ is reducible in $\mathbb{Z}[x]$. It seems $n=4k$ is the only one (the factorization follows easily from Sophie Germain's identity in this ...
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2answers
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Polynomial quotient rings

I have a quotient ring $R=\Bbb Z[t]/(1-t)^3$. It is asked to show that $\overline{2t^3-2}=\overline{6t^2-6t}$ and $\overline{1-4t^3}=\overline{4t-6t^2-t^4}$. I feel like I am missing some important ...
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Matrix representation of a linear operator

As I'm studying for my final, my book keeps skipping alot of steps and I don't know how tthey get from point a to point b - probably because its elementary at that stage in the book, except not to me ...
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The ring of fractions $K(x)$ is the field generated by $K$ and $x$.

I would like to show that the ring of fractions $K(x)$ of $K[x]$ in an extension $L$, where $K\subset L$ fields, is the field generated by $K$ and $x$ (let's call it by $\tilde{K(x)}$). I know just ...
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Rational functions are decomposed in polynomial products

I'm trying to understand why this is true: Since $K(x)$ is a field, $K(x)$ is an UFD, then $K(x)$ can be written uniquely as products of irreducible elements of $K(x)$. I didn't understand why ...
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How prove such $f(x)g(x)$ coefficient is $1$ or $-1$

show that: there are exsit two polynomial $f(x),g(x)$ with the coefficient is integer,the polynomial $f$ and $g$ the coefficient at least one more than $2014$,and $f(x)g(x)$ coefficient is ...
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1answer
43 views

Is it possible to calculate the roots of the difference between successive terms of this polynomial series $\rm{P}_n (x)=x\rm{P}_{n-1}-r\rm{P}_{n-2}$

Consider the polynomial series defined by the following recursion formula: $$ \begin{align} &\mathrm{P}_0 = 1 \\ &\mathrm{P}_1 = x-r \\ &\mathrm{P}_n = x\mathrm{P}_{n-1} - ...
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3answers
112 views

reduction of $6xy + 8 y^2 -12x-26y + 11 = 0$ to canonical one of a second-order curve

I have this polynomial $$ 6xy + 8 y^2 -12x-26y + 11 = 0 $$ and I need to reduce it to a canonical equation of a second-order curve. The correct answer from the textbook is that it is a hyperbola ...
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4answers
81 views

Solve the following equation: $\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$

Solve the following equation: $$\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$$ I know it's from a Math Olympiad but I don't know which and I couldn't find it on the internet. Expanding everything ...