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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47 views

Sum of $p$th powers using polynomial interpolation

It is well known that the sum of the first $n$ $p$th-powers is polynomial in $n$ and is given by: $$ \sum_{k=1}^n k^p = \frac{1}{p+1} \sum_{j=0}^p (-1)^j {p+1 \choose j} B_j n^{p+1-j} $$ where $B_i$ ...
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1answer
43 views

I have a question about Viete's formulas

If I have a polynomial $a_n x^n + a_{n-1}x^{n-1}+ \cdots + a_1 x + a_0$, and the roots of the polynomial is $r_1,r_2,\ldots,r_n$, then I can rewrite the polynomial as, $a_n x^n + a_{n-1}x^{n-1} ...
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3answers
38 views

$x^2+3$ has two zeros over ${\Bbb F}_p$ provided that $x^2+x+1\in{\Bbb F}_p[x]$ has two?

The following is an exercise in abstract algebra: If $p=1\pmod{3}$, then $x^2+x+1\in\Bbb{F}_p[x]$ has two zeros. Prove in this case that $-3$ is a quadratic residue mod $p$. Showing that ...
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0answers
37 views

Kneser Inequality in multivariables

Based on the Kneser Inequality ("Polynomials and Polynomial Inequalities", p. 260) one has $\Vert q \Vert_{[-1, 1]} \Vert r \Vert_{[-1, 1]} \leq C(n, m) \Vert q r \Vert_{[-1, 1]}$ where all norms ...
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1answer
81 views

Polynomials, prove exercise question about question

There is a polynomial P with integer coefficients and with pairwise different integers $a,b,c$ . Prove that it is not possible for $P(a) = b$, $P(b)=c$, $P(c) = a$ First off I don't understand ...
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1answer
79 views

Factoring polynomial $x^3−2x^2−4x−8$ that fails Bezout's identity test

I usually factor 3rd degree polynomial in two steps. First, I find all the divisors of the last, coefficient-free part of the polynomial (in this case that's 8) and try (applying Bezout's identity) to ...
3
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1answer
63 views

Polynomial with n real roots

Let $P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_{1}x + 1$ where $a_i$ are nonnegative and real. Assume $P$ has $n$ real roots. Prove $P(2) \geq 3^n$. I thought I had a good idea about ...
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1answer
62 views

Taylor theorem for f(x+h)

I am following a proof that applies Taylor's theorem on this document (http://www.gautampendse.com/software/lasso/webpage/pendseLassoShooting.pdf) I am not understanding one of the terms explained on ...
2
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1answer
25 views

Find $a$ and $b$ such that $g$ divides $f$ evenly

$f=2X^4-3X^2+aX+b,\ g=X^2-2X+3, \ f,g \in \mathbb{Q}[X]$ I have tried to divide $f$ by $g$ but I get $ (a+10)X +b +3$ as the remainder which looks like a bad result. I have, also, tried to factor ...
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1answer
44 views

Definition of monomial

I thought the definition of a monomial is an algebraic term that has no subtraction or addition. I saw on my online college homework that 2/x is not a monomial. Why?
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3answers
57 views

Can every polynomial be factored into constant and linear complex factors?

That is, can any polynomial, $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x^1+a_0$, be expressed $b_0\left(x + b_1\right)\left(x + b_2\right)\ldots \left(x + b_n\right)$ where $b_i \in \mathbb{C}$?
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0answers
69 views

Efficient Computation of Swinnerton Dyer Polynomials

the Swinnerton-Dyer polynomials are defined as $$SD_n(x) = \prod(x \pm \sqrt{2} \pm \sqrt{3} \pm ... \pm \sqrt{p_n})$$ where the product is taken over all possible permutations of $+$ or $-$ signs. ...
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4answers
64 views

Can I find solutions to $a^4 + a^2 + a = b^2 + b$, $a,b \in \mathbb{Z}$ and $ 1 < a < b$?

I was wondering if anyone could point me in the correct direction for either finding a solution to my problem or proving that it does not exist. $$a^4 + a^2 + a = b^2 + b \;\text{ for }\; a,b \in ...
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1answer
50 views

How can I prove this about the tangent line formula??

The equation of a tangent line to $f(x)$ at $x = t$ is $y = f'(t)(x - t) + f(t)$. Recently, I heard that it is also determined by the remainder of polynomial division of $f(x)$ by $(x-t)^2$. For ...
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1answer
114 views

CHKMO 2015 and cubic equations

Let $a,b,c$ be distinct real numbers. If the equations $E_1: ax^3+bx+c=0, E_2: bx^3+cx+a=0$ and $E_3: cx^3+ax+b=0$ have a common root, prove that at least one of these equations has three real ...
3
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1answer
38 views

Polynomials with purely imaginary coefficients?

Finished a homework problem concerning polynomials with all real coefficients and why complex roots of p(z)=0 come in pairs. Curious is there is a similar situation for polynomials with all purely ...
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1answer
66 views

Does Euclidean division not work for general polynomials?

If $K$ is a field. Then in $K[X]$ there is an Euclidean algorithm and if $K$ is replaced by any arbitrary commutative ring $R$, then almost we have an Euclidean algorithm, by the following result: ...
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4answers
162 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.
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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 ...
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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...
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2answers
65 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 $k$ ...
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6answers
573 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 ...
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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!
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1answer
94 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 - ...
12
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1answer
168 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 ...
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1answer
42 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: $-2x^1$. ...
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4answers
154 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.
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1answer
189 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?
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1answer
95 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 ...
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1answer
151 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 ...
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4answers
72 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 ...
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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 ...
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1answer
603 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 ...
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1answer
73 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 ...
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2answers
45 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 ...
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1answer
57 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 ...
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2answers
46 views

Factorization of a Polynomials

Does Mathematical induction work?
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4answers
90 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!
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1answer
51 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 ...
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2answers
47 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 ...
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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) → ...
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1answer
164 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 ...
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3answers
122 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. ...
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3answers
69 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
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1answer
237 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
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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 $$ ...
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1answer
31 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, ...
0
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1answer
30 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
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5answers
266 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
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1answer
84 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 ...