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.

learn more… | top users | synonyms

0
votes
4answers
205 views

How many solutions has this third degree equation?

how many solutions has this equation: $$ {x}^{3}+4\,{x}^{2}-1=0 $$ i tried ruffini so far and it is not working, now i'm stuck and no idea of how to aproach this.
1
vote
1answer
35 views

Two sets of polynomials with distinct roots build the ring of polynomials.

Definitions: $i \in K$ $U_{i}:=\{f\in K[X] |f(i)=0 \}$ $K[X]$ is the ring of polynomials HINTS: K[X] is a vector space Every $U_{i}$ is a vector subspace of $K[X]$ Question: (i) With $s \neq ...
0
votes
1answer
22 views

Does the equality $\partial^\alpha(x^\alpha)(0)=\alpha!$ hold?

Do we have $\partial^\alpha(x^\beta)(0)=\alpha!=\beta!$ if $\alpha=\beta$ and $0$ else? I tried to proof it on induction, can include my attempts if needed, but they seem to have failed anyway...
1
vote
2answers
68 views

A polynomial with integer coefficients that attains the value $5$ at four distinct points

There is a polynomial $f$ of integer coefficients such that $\deg(f) \geq 4$. Let's assume that there are four integers $a,b,c,d$ for which $f(a)=f(b)=f(c)=f(d)=5$. Prove that there is no integer $k$ ...
4
votes
6answers
676 views

Prove that $x-1$ is a factor of $x^n-1$

Prove that $x-1$ is a factor of $x^n-1$. My problem: I already proved it by factor theorem† and by simply dividing them. I need another approach to prove it. Is there any other third approach ...
1
vote
2answers
67 views

transformation of $y=3(4-x)^3-6$

I am looking for the expansion of $y=3(4-x)^3-6$. I got confused about the $(4-x) $ part. Please help, thanks!
1
vote
1answer
106 views

Polynomial rings, division algorithm

Let $m,n$ be non-negative integers and $m>n$. Find polynomials $g(x),r(x)$ from the ring $R[x]$ such that $x^m -1 =q(x)(x^n-1) + r(x)$ , $r(x)=0$ or $\deg(r(x))<n$. In which case $x^n -1|x^m - 1$...
12
votes
1answer
173 views

$P(z)=0$ iff $Q(z)=0$, $P(z)=1$ iff $Q(z)=1$. Prove that $P(x)=Q(x)$ for all $x$

Assume $P(x)$ and $Q(x)$ are polynomials with complex coefficients with degree greater than or equal to $1$ such that $P(z)=0$ if and only if $Q(z)=0$, $P(z)=1$ if and only if $Q(z)=1$. Prove that $P(...
0
votes
1answer
60 views

Do monomials' degrees always depend on the whole-number exponent of the variable or whether it's a constant (having a degree of zero)?

Is it true that the monomial $4x^4$ has a degree of $4$ because of the exponent? Also, I think $-2x$ has a degree of $1$ because it has an exponent of $1$ when it's also written like this: $-2x^1$. ...
10
votes
4answers
159 views

Polynomial equation $f(x)f(2x^2)=f(2x^3+x)$

Find all polynomials $f(x)$, for which $f(x)f(2x^2)=f(2x^3+x)$. I have no idea how to do it.
6
votes
1answer
207 views

If all the roots of a polynomial P(z) have negative real parts, prove that all the roots of P'(z) also have negative real parts

If all the roots of a polynomial $P(z)$ have negative real parts, prove that all the roots of the derivative $P'(z)$ also have negative real parts. Could anyone provide a proof for this please?
2
votes
1answer
109 views

Chevalley's theorem proof

I'm trying to prove Chevalley's theorem stating that $$ \text{If } f \in \mathbb{Z}[x_1, \dots, x_n] \text{ is a form of degree } r < n \text{,}$$ $$ \text{then there exists a nonzero solution of }...
4
votes
1answer
155 views

Why is 105th cyclotomic polynomial interesting?

According to Wikipedia the $105$th cyclotomic polynomial is interesting because $105$ is the lowest integer that is the product of three distinct odd prime numbers and this polynomial is the first ...
1
vote
4answers
80 views

Polynom equality modulo p

I found these two equations: (a) $$X^4 + 1 \equiv (X + 1)^4 \mod \ 2$$ (b) $$X^4 + 1 \equiv (X^2 - X - 1)(X^2 + X - 1) \mod \ 3$$ I would like to understand the concept of modulo for Polynoms. How ...
3
votes
2answers
40 views

Polynomial $(x − a)^2(x − b)^2 + 1$ is not the product of two polynomials with integral coefficients

Let $a, b$ be integers. Then the polynomial $(x − a)^2(x − b)^2 + 1$ is not the product of two polynomials with integral coefficients. Suppose $(x − a)^2(x − b)^2 + 1 = p(x)q(x)$ then $p(a)q(a)=1$,...
3
votes
1answer
778 views

Irreducible polynomials over the reals

Everybody knows that the degree of irreducible polynomials over the reals is either one or two. Is it possible to prove it without using complex numbers? Or without using fundamental theorem of ...
1
vote
1answer
79 views

Roots less than 1 if at least one coefficient is greater than one

I have this doubt. If you have this equation with $\alpha_i \in \mathbb R$ $$P(z)=1-\alpha_{1}z-\alpha_{2}z^{2}- \cdots - \alpha_{p}z^{p}=0$$ I believe that if there exist an $\alpha$ greater or equal ...
3
votes
2answers
47 views

Prove relations between the roots of 3 quadratic equations

Let $x_1, x_2$ be the roots of the equation $x^2 + ax + bc = 0$, and $x_2, x_3$ the roots of the equation $x^2 + bx + ac = 0$ with $ac \neq bc$. Show that $x_1, x_3$ are the roots of the equation $...
2
votes
1answer
58 views

Polynomial prove exercise

$P(x)=x^n + a_1x^{n-1} +\dots+a_{n-1}x + 1$ with non-negative coefficients has $n$ real roots. Prove that $P(2)\ge 3n$ I don't have an idea how to do that, I'm in 4th grade high school, you don't have ...
1
vote
2answers
46 views

Factorization of a Polynomials

Does Mathematical induction work?
1
vote
4answers
91 views

What is the remainder when $x^7-12x^5+23x-132$ is divided by $2x-1$? (Hint: Long division need not be used.

What is the remainder when $x^7-12x^5+23x-132$ is divided by $2x-1$? (Hint: Long division need not be used.) The Hint is confusing!
-1
votes
1answer
60 views

What is the minimum degree for a curve that has two different points.

I'm having some difficulty solving this problem. The information I have is the following: What is the minimum degree for a curve that has two different points.( 2 different ordered pairs let s say (...
1
vote
2answers
49 views

Prove that$a^2+b^2$ is composite from the information provided.

Suppose $\alpha$,a,b are integers and $b\neq-1$. Show that if $\alpha$ satisfies the equation $x^2+ax+b+1=0$,then prove $a^2+b^2$ is composite. I am starting with this study course of polynomials and ...
1
vote
1answer
30 views

Is the taylor polynomial of degree $2$ near $(0,0)$ of $𝑓(𝑥, 𝑦) = \frac{1}{ 2 - (𝑥 + 𝑦^2)}$ the following:

$ P(𝑥, 𝑦) = \frac{1}{2} + \frac{𝑥}{4} + \frac{𝑥^2}{4} + \frac{𝑦^2}{2}$ Is this right? I can't tell, as I can't seem to see the remainder going to $0$ when divided by $x^2 + y^2$ as $(x, y) → (0,...
7
votes
1answer
170 views

Can $f(g(x))$ be a polynomial?

Let $f(x)$ and $g(x)$ be nonpolynomial real-entire functions. Is it possible that $f(g(x))$ is equal to a polynomial ? edit Some comments : I was thinking about iterations. So for instance $f(f(...
1
vote
3answers
129 views

How do I show that the polynomial $f(x) = x^2 + x + 3$ $∈$ $Z_7[x]$ is a primitive polynomial?

I understand that a primitive polynomial is a polynomial that generates all elements of an extension field from a base field. However I am not sure how to apply this definition to answer my question. ...
0
votes
3answers
72 views

Will someone explain this polynomial regression equation?

I am in high school and I need to write a program that does polynomial regression to any degree on a set of data for a personal project. I think that this Wikipedia Article has the equation that I ...
4
votes
1answer
246 views

How to solve 29 coupled quadratic equations?

I have a set of 29 coupled quadratic equations, with 29 unknown variables. Can anyone offer any advice on how I could go about solving this? 3 days of staring at a wall has so far given me no ...
4
votes
1answer
79 views

Can the natural proof of this algebraic identity be simplified?

Let $x^4+c_3x^3+c_2x^2+c_1x+c_0$ be a real polynomial with no real root. Then there are two pairs of conjugate complex roots, $a_1\pm b_1 i$ and $a_2\pm b_2 i$, and one has the identity $$ c_1^2-...
1
vote
1answer
32 views

Bézout's identity on a polynome sequence

I'm stuck on an exercise which is split in 3 questions : 1) Prove that : $$\exists (U_n, V_n) \in \mathbb{R}[X]^2 \text{ s.t. } (1-X)^{n+1}U_n+X^{n+1}V_n=1$$ 2) Let $(R_n, S_n)\in\mathbb{R}[X]...
0
votes
1answer
31 views

Total number of distinct solution produced by polynomial

I have a function $F(x,y) = ax + by$ where $x,y$ belongs to range $[1..10^{10}]$ and $a$ and $b$ are constants, all are integers. How many distinct values can be produced by this function, please give ...
4
votes
5answers
282 views

Curious Binomial Coefficient Identity

Consider the following set of identities: ${m+1\choose 1}={m\choose 1}+1$, ${m+1\choose 2}=2\binom m 2 - {m-1\choose 2}+1$, ${m+1\choose 3}=3\binom m3-3{m-1\choose 3}+{m-2\choose 3}+1$, ... This set ...
2
votes
1answer
86 views

On the proof of Fejér-Riesz theorem

I'm having a course about Analytic Number Theory, and I'm having trouble understanding the proof of Fejér-Riesz Theorem: http://people.virginia.edu/~jlr5m/Papers/FejerRiesz.pdf First of all, I didn't ...
2
votes
1answer
266 views

Solving polynomial equation system to find three dimensional location

For an embedded systems project, I need to solve a system of equations. However, my algebraic skills are limited, and I am not able to solve it. This question consists of the following parts. The ...
2
votes
1answer
113 views

polynomial approximation - basic chebyshev question

I was asked to find the best linear approximation to $f(x)=x^2$ in $x \in [0,1]$ using chebyshev polynomials, meaning, using the known property that $2^{1-n}T_n(x)$ is the best approximation to $0$ at ...
2
votes
1answer
65 views

Is $(X^3 - 18X + 12, 5) \in \mathbb{Z}[X]$ a prime ideal?

I'm trying to determine wheter $A = (X^3 - 18X + 12, 5)$ and $B = (X^3 - 18X + 12, X-1)$ is a prime ideal in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. I know that $A = \mathbb{Q}[X]$ since I can make $\...
2
votes
1answer
159 views

How do I find the coefficient of $x^2$ in this polynomial, given its value at three points?

We are given the following data about a polynomial $P(x)$ of unknown degree: $P(0) = 2$, $P(1) = −1$, $P(2) = 4$. Determine the coefficient of $x^2$ in $P(x)$ if all the third-order differences are $...
2
votes
1answer
103 views

Elementary proof of the irreducibility of $T^4 - a T - 1$ in $\mathbf{Q}[T]$ when $a\in\mathbf{Z}-\{0\}$

This is from the exercises of Bourbaki, Algèbre, Chapitre V, first exercise of the exercises concerning the second paragraph of the fifth chapter. (p. 140.) As Gauss Lemma ("if your gcd is ...
2
votes
1answer
81 views

Proof $27X^3 - 13X^2 + 180 \in \mathbb{Q}[x]$ is irreducible without Gauss's lemma

I'm asked to show that $27X^3 - 13X^2 + 180 \in \mathbb{Q}[x]$ is irreducible in $\mathbb{F}_{13}[X], \mathbb{Z}[X]$ and $\mathbb{Q}[X]$. I've managed to proof the first two. In $\mathbb{F}_{13}[X]$ I ...
0
votes
2answers
35 views

Polynomial division with modulo

The following polynomials are in the field of whole numbers $mod$ $5$, so $x =5x$. $f(x)=2+4x⁴+4x⁵+x⁶$ $g(x)=3+x+4x²+x³$ $f=q*g+r$ Is there a solution where the degree of $r$ is smaller than the ...
0
votes
1answer
178 views

Is reducing factoring of integers to finding a polynomial which takes a perfect square value useful?

Below we only consider numbers $N=pq$, where $p$,$q$ are primes of the form $ (6j+1)$. It's easy to show that $N + 9n^2 = d^2$. $9k^2$ is related to $(p-q)$ and $d^2$ is related to $(p+q)$. the $n$ in ...
0
votes
1answer
322 views

Newton's forward-difference formula question?

Use Newton's forward-difference formula to construct interpolating polynomials of degree two, and three for the following data. Approximate the specified value using each of the polynomials. I ...
1
vote
0answers
140 views

Different ways to prove Fundamental Theorem of Algebra

This is just a curosity .I know some proofs of the fact that Every non constant polynomial with complex coefficient has a complex root via using Liouville's theorem in Complex Analysis.Proof goes as ...
1
vote
1answer
59 views

To find nature of roots of $Ax^{4} + Bx^{3} + Cx^{2} + Dx - E $

To find nature of roots of $$f (x) = Ax^{4} + Bx^{3} + Cx^{2} + Dx - E $$ Where $A, B, C, D, E$ are all positive. After applying Descartes' Rule of signs to $f(x)$ there is one sign change , so ...
0
votes
1answer
14 views

Suppose that $p(x)=1/4x^4−2/3x^3-5/2x^2+6x-1/12 $withDom(p)=[1,2].Define$q(x)=p^−1(x)$. Show, algebraically, why q(x) exists

I don't know where to start. What does it means to define $q(x) = p^-1(x)$?
3
votes
4answers
124 views

Is the zero polynomial the only polynomial that vanishes at every point of $\mathbb C$?

The zero polynomial has the property that every value it takes on $\mathbb C$ is zero. Is the converse true, or are there other polynomials $f$ such that $ f(x)=0$, for all $x \in \mathbb{C}$?
1
vote
0answers
46 views

Mathmatical notation for a term of a polynomial

If I have a polynomial $f(x) = ax^n + bx^{n-1} + cx^{n-2} \ldots zx^0$, is there any mathematical notation for one term, such as the $x^3$ term. For example, if I have a polynomial of $f(x) = x^6 + \...
0
votes
1answer
51 views

Explicit form for $\left(e^{-x^2}\left(\frac{d^n}{dx^n}e^{x^2}\right)\right)^2$

Basically I have been working with polynomials of the form: $$P_n(x)=e^{-x^2}\left(\frac{d^n}{dx^n}e^{x^2}\right)$$ I do realize that an explicit form for $P_n(x)$ has been asked for on this site ...
1
vote
4answers
111 views

If $x^2+x+1=0 $, then find the value of $(x^3+1/x^3)^3$?

If $x^2+x+1=0 $, then find the value of $(x^3+1/x^3)^3$ This thing doesn't make sense how should I use first identity to find the second one.
1
vote
1answer
37 views

What is the series to converge with $1/x$ from $(1,\infty)$?

I'm trying to find an alternative series of polynomials that can pssibly converge with $\frac{1}{x}$. So far I know that the taylor series for $\frac{1}{x}$ is, as should be known, $$\sum_{n=0}^{\...