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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Rational representation of conics

Currently I'm beginning my study of rational curves (Rational Bezier and NURBS) all books that I've read tell me that is "well known" that conics can't be represented by Bezier or even a B-Spline. ...
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Coefficients of the polynomials generated by $f_0=x,\ f_{i+1}=f_i\pm\dfrac1{f_i}$.

Let $f_0=x,\ f_{i+1}=f_i\pm\dfrac1{f_i}$ for $i\geq0$, i.e., $f_i=\dfrac{\sqrt{f_{i+1}^2\mp4}+f_{i+1}}2$ I have observed that $f_1=\dfrac{x^2\pm1}x$ $f_2=\dfrac{x^4\pm3x^2+1}{x(x^2\pm1)}$ ...
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1answer
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Is $Y=aX^b\cdot\exp(X)$ a rational or exponential function?

Is $Y=aX^b\cdot\exp(X)$ a rational or exponential function? $Y$ and $X$ are real variables, $a$ and $b$ are parameters. Someone said this is a product of polynomial and exponential function. Do we ...
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2answers
29 views

If I have a polynomial $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ with a solution at $x = -1$, how do I get the other root

If I have a polynomial $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ with a solution at $x = -1$, then I know I can just take $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ and divide it by $x+1$ to get the other root. In a ...
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37 views

Algebraic relations between trigonometric numbers

Given $n\in2\Bbb N$, what is precise algebraic relation between $cos\frac{\pi}{n-1}$,$cos\frac{\pi}{n+1}$? Both numbers are algebraic, which implies there should be an algebraic relation between ...
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Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors?

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors? For example: $f=x^5+3x^4+x^3+4x^2+1$, and $g=x^5+3x^4+4x^3+3x+1$ Can we represent these ...
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11 views

Cleaning Up Messy Product Notation

Suppose I have the following: Let $N_1<...<N_m$. Let $T_{N_k}(x)=\sum_{i=0}^{N_k}{\frac{x^i}{i!}},$ $ t(i,j,x)=T_{N_i}(x)-T_{N_j}(x)$ I'm trying to define a polynomial $p_k(x)$ like this: ...
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38 views

Expression for Taylor's formula with a remainder

Assume $f$ has a continuous second derivative $f~''$ in some neighborhood of $a$.Then, for every $x$ in this neighborhood, we have $f(x) = f(a) + f~'(a)(x-a) + E_1(x)$ , where $E_1(x) = \int_a^x ...
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What's the difference between these two definitions of polynomial function?

Definition 1: Given $a_n,...,a_1,a_0 \in \mathbb{R}$, a polynomial function is a function $p:\mathbb{R} \rightarrow\mathbb{R} $ such that $p(x)=a_nx^n+...+a_1x+a_0$ Definition 2: The function ...
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polynomial and rational functions equation, find the values of a and b that will make the statement true

Find the value of $a$ and $b$ to make the given condition true: $ax^3-bx^2+45x+54$ has $3$ as a zero and yields a remainder $12$ when divided by $x+1$
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14 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
23 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
31 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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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
32 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
33 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 ...
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50 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
17 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 ...
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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
15 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
35 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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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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23 views

Polynomial “factorization” on a set of polynomials

Considering a set of multivariates polynomials $\{P_1(x,y,...),...,P_n(x,y,...)\}$, I wish to know if a given $P(x,y,...)$ could be expressed as a sum of products of $P_i$, such as ...
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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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29 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
24 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 ...
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24 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
47 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
63 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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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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18 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
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polynomial of $4^\text{th}$ degree, prove

There is a polynomial $f$ of integer coefficients such that $\text{deg(f)} \geq 4$. 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 ...
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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
58 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
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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 $x^n -1|x^m - ...
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$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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I need help to solve this problem [duplicate]

Let $m,n$ be negative integers and $m>n$. Find polinomials $g(x),r(x)$ 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 ...
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1answer
13 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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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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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
25 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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2answers
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Why is 105th cyclotomic polynomial interesting?

According to wikipedia the 105th cyclomatic 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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36 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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>Find all pairs of positive integers $(m, n)$, so that $1 + x + x^2 +\ldots+x^m \mid 1 + x^n + x^{2n} +\ldots +x^{mn}$

Find all pairs of positive integers $(m, n)$, so that $1 + x + x^2 +\ldots+x^m \mid 1 + x^n + x^{2n} +\ldots +x^{mn}$ I have to find $(m, n)$ such that ...
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2answers
35 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
73 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
44 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
35 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
43 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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18 views

Factorise the following polynomial

file://localhost/var/folders/0p/frxrkc9d4_z99dy684t4_9100000gn/T/LaTeXiT-2.6.1/latexit-drag.pdf How do you factorise the above?