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.

learn more… | top users | synonyms

1
vote
2answers
30 views

Why are the coefficients always equal?

Take the equation $ax^{2} + bx + c = 3x^{2} + 4x + 53$. Why is it always true that $a = 3, b = 4$ and $c = 53$? I've seen many examples like this where the coefficients are equated, and was just ...
0
votes
0answers
7 views

Measure of variation(?) of multidimensional polynomial function

I have a multidimensional function $$\mathbf{f}(x) = [f_0(x), ... , f_N(x)]$$ where $f_n$ are real-valued trigonometric polynomials$. I want to measure how much $\mathbf{f}(x)$ varies over some ...
1
vote
3answers
21 views

Constructing Polynomial Function from Set of Points and Slopes

I only have a basic knowledge of calculus but I would like to know if it's possible to, given a set of points each with their own slopes, construct the simplest (or any) polynomial function that ...
0
votes
3answers
60 views

Polynomial whose one of its roots is $\cos(\pi/7)$

Let $P(x)$ be a 3rd-degree polynomial with integer coefficients, one of whose roots is $\cos(\pi/7)$. Compute $\frac{P(1)}{P(-1)}$ I saw this question in a contest math problem, and I know that it ...
-1
votes
5answers
63 views

Let $f(x)$ be polynomial of degree four [on hold]

Let $f(x)$ polynomial of degree four where: $$f(1)=1,f(2)=4,f(3)=9,f(4)=16, f(7)=409$$ Find $$f(5)=??$$
-4
votes
4answers
62 views

Suppose that $\alpha$ root of the equation [on hold]

Suppose that $\alpha$ root of this equation: $$x^4+x^2-1=0$$ Find the value of $$\alpha ^{6}+2\alpha ^{4}$$ "I want the way, not the roots of the equation." I tried, but I couldn't find any thing.
0
votes
0answers
6 views

Trace of an element in a separable field extension

Let $L=K(\alpha)$ be a finite separable field extension of $K$ of degree $n$ and let $\alpha$ have minimal polynomial $f(X)\in K[X]$ with roots $\alpha=\alpha_1,...,\alpha_n$. Write ...
11
votes
3answers
560 views

Are polynomials infinitely many times differentiable?

Are polynomials infinitely many times differentiable? If so, does it only mean that at some point we reach 0 and then we keep on getting 0? Thank you!
2
votes
1answer
21 views

Example of $Q((x))$ that doesnt match field of fractions of ring $F[[x]]$

Let $F$ be a commutative ring without zero divisors and $Q$ -its field of fractions. Let $Q(x)$ be also field of fractions of ring $F[x]$. How can field $Q((x))$ not match field of fractions of ring ...
0
votes
2answers
26 views

Three polynomials as unknowns of an equation

If three polynomials $f,g,h\in\mathbb R[x]$ are such that $[f(x)]^2 –x[g(x)]^2+[h(x)]^2=0$, what can we conclude about $f, g, h$?
0
votes
2answers
29 views

Find the sum of the roots of the exponential equation

The equation $$2^{333x - 2} + 2^{111x + 2} = 2^{222x + 1} + 1$$ has three real roots. Find their sum. I'll simplify it first as: $$\frac{1}{4}2^{333x} + (4)2^{111x} = (2)2^{222x } + 1$$ Let ...
2
votes
3answers
49 views

Find the sum of the roots given no multiple roots.

Find the sum of the roots, real and non-real, of the equation $$ x^{2001} + \left( \frac{1}{2} - x \right)^{2001} = 0 $$ given that there are no multiple roots. I am in a weird situation here. ...
6
votes
3answers
110 views

How can I prove irreducibility of polynomial over a finite field?

I want to prove what $x^{10} +x^3+1$ is irreducible over a field $\mathbb F_{2}$ and $x^5$ + $x^4 +x^3 + x^2 +x -1$ is reducible over $\mathbb F_{3}$. As far as I know Eisenstein criteria won't ...
1
vote
3answers
40 views

Multiplicity of a root of a polynomial

:) It's true that, if a polynomial has a root (let's say, k, for example) with multiplicity n (n>1, for n integer), then it's true that the derivate polynomial have k as a root with multiplicity ...
1
vote
1answer
60 views

A polynomial that satisfies $x^pf(1-x) + (1-x)^pf(x) = 1$

The context of this question is the construction of the Daubechies wavelet. $f$ is a polynomial of degree $p-1$ which satisfies the equation: $$ x^pf(1-x) + (1-x)^pf(x) = 1 \tag{1} $$ Since $$ ...
0
votes
2answers
32 views

Proof of associativity of polynomials product (infinite variables)

The product of polynomials in $R[X_i]_{i\in I}$ where $I$ is not necessarily finite is associative ($R$ commutative ring), but I can't find any detailed proof of this fact. Either it is left in ...
0
votes
1answer
13 views

module isomorphism inbetween two equivalence classes of polynomials

Let $g \in \mathbb{R}[t]$ be a normed irreducible polynomial of degree 2, meaning that $g(t) = (t - \lambda)(t - \overline{\lambda}$) for a $\lambda = a + b i$, with $a, b \in \mathbb{R}$, $b ≠ 0$. I ...
-3
votes
2answers
39 views

Polynomial division in the case of $\frac{x^2 -x}{1-x}$ [on hold]

What is the answer in $$\frac{x^2 - x}{1-x}$$
6
votes
2answers
72 views

Proving that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$.

I need to prove, that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$. Prove that $$x^m+x^{-m}=P_m (x+x^{-1} )=a_m (x+x^{-1} )^m+a_{m-1} (x+x^{-1} )^{m-1}+...+a_1 (x+x^{-1} )+a_0$$ on ...
2
votes
0answers
6 views

Special class of Brenke Polynomials

I was wondering if there are any particular papers dealing with a particular class of Brenke Polynomials, defined as $$A(t)B(xt)=\sum_{n\ge 0}P_n(x)t^n$$ where $A=B$ or, where $A(t)=C(B(t),t)$ for a ...
3
votes
1answer
148 views

What is the minimum degree of a polynomial for it to satisfy the following conditions?

This is the first part of a problem in the high-school exit exam of this year, in Italy. The differentiable function $y=f(x)$ has, for $x\in[-3,3]$, the graph $\Gamma$ below: $\Gamma$ exhibits ...
6
votes
1answer
63 views

When proving that f(z) is a polynomial, is it enough to consider just one point instead of keeping z arbitrary?

I think so - but I'd rather ask the MSE community too. Say I am given the bound |f(z)| < $|z|^3$, and that f is entire. Show f must be a polynomial. I used Cauchy's Integral Formula for ...
1
vote
1answer
22 views

Evaluation of polynomials at tensor products

Let $S,T$ be $R$-Algebras, $f \in S[X]$ a polynomial. in my notes it says you can easily lift $f$ to a ploynomial $f'$ in $(S \otimes T)[X]$. But I have no idea what $f'(s \otimes t)$ is. My guess is ...
-1
votes
2answers
49 views

What is the remainder when a polynomial $g(x^{12})$ is divided by $g(x)$? [on hold]

Let $g(x) = x^5 +x^4 +x^3+x^2+x+1$. What is the remainder when the polynomial $g(x^{12})$ is divided by the polynomial $g(x)$?
4
votes
2answers
41 views

Degree of minimal polynomial

The minimal polynomial of $a$ over $\mathbb{Q}$ is quadratic. The minimal polynomial of $b$ over $\mathbb{Q}$ is cubic. Is the minimal polynomial of $a+b$ necessarily of degree $6$? If so, what is ...
1
vote
1answer
25 views

Find parameter m if equation admits three distinct real solutions

$2x^3+3x^2-x+5-m=0$ I know for the above equation there is the following condition for the case when all the three roots must be distinct and real: $D = -4b^3d + b^2c^2 - 4 ac^3 + 18abcd - 27a^2d^2 ...
0
votes
0answers
21 views

Find Next Position and Velocity from Instantaneous Values

To find the position of an object at a given point in time: $y_0 + v_0t - \frac{32t^2}{2} = y_t$. And to find the object's speed at a given point in time: $v_0 - 32t = v_t$ So say I give the ...
3
votes
0answers
71 views

What does it actually mean by a “Characteristic Polynomial”?

Please can you describe in layman's term, what does it actually mean by a "Characteristic Polynomial"? Is it a property only of Matrices? What does it describe about a Matrix, that is, what can we ...
0
votes
2answers
39 views

Remainder of division.

What's the remainder of dividing a polynomial $P(x)=x^{2008}+x^{2007}+1$ with binomial $x^2+1$. It has to be: $$x^{2008}+x^{2007}+1=(x^2+1)Q(x)+(Ax+B)$$ But when substituting variable $x$ with a ...
1
vote
0answers
25 views

Zeros of derivative of composition of polynomials

Let $f(x),g(x)$ be polynomials such that their derivatives $f'(x),g'(x)$ have $n$ and $m$ real roots. What is the possible minimal/maximal numbers of real roots for the polynomial $(f(g(x))'$? My ...
-4
votes
0answers
9 views

State and proof Division Algorithm for polynomials. [closed]

State and prrof Division ALgorithm for Polynomials.
2
votes
3answers
36 views

Determine roots of a polynomial with variable exponent

I need to know the nature of the roots of the equation $$ x(x+a)^b -1 = 0 $$ when changing a and b, where $ a,b $ are natural numbers, I've looked around on the web but I was unable to find how to do ...
1
vote
1answer
23 views

When are monic polynomials of fourth degree divisible?

Note that this might be an X/Y problem, therefore I'm posting the original question too. I am asked to prove that given a monic polynomial of fourth degree which has a non-zero root, must have at ...
2
votes
1answer
54 views

Product of integer polynomials has coefficients $0,\pm 1$

Let $n$ be a positive integer. Do there always exist for any $n$ two polynomials $P(x),Q(x)$ with integer coefficients such that both $P(x),Q(x)$ have some term with coefficient greater than $n$ in ...
0
votes
1answer
26 views

Remainder of polynomial division.

Remainder of dividing a polynomial $P(x),$ $ \left (\deg{P(x)\geqslant2} \right ) $ with $(x-1)$ is $1$ while remainder of dividing the same polyinomial with $(x+1)$ is $-1$. Find the remainder of ...
2
votes
2answers
23 views

Property of polynomials proof

Let$$P(z)=\sum_{k=0}^n a_kz^k=a_0+a_1z+...+a_nz^n$$ be an N-th degree polynomial of a complex variable z, where the $a_k$ are complex constants. Now,$$\vert a_0\vert-\vert a_1\vert x-...-\vert ...
3
votes
0answers
51 views

Resultant of two polynomials in two variables

I have two polynomials in two variables. $$f= nx^n+(n-1)x^{n-1}y+(n-2)x^{n-2}y^2+...+xy^{n-1}-c$$ $$g= x^{n-1}y+2x^{n-2}y^2+3x^{n-3}y^3+..+(n-1)xy^{n-1}+ny^n-d$$ Where $c$ and $d$ are some ...
1
vote
2answers
57 views

Prove that field $Q(x)$ is a field of fractions of ring $F[x]$

Let $F$ be a commutative ring without zero divisors and $Q$ its field of fractions. How can I prove that field $Q(x)$ is a field of fractions of ring $F[x]$? And also why is it that field $Q((x))$ ...
3
votes
3answers
48 views

Monic polynomial $= 0 \mod p$ for all $x$

For a monic polynomial with integer coefficients (leading coefficient of $1$) $f(x)$ where $f(x) \equiv 0$ mod $p$ for all $x$, where $p$ is a prime number how do I show that the degree of the ...
0
votes
2answers
86 views

How to factorise $x^4$ equations?

This is my previous question I'm facing a problem to factorise this $64x^4+64x^3-88x^2-51x+39=0$. How to factorise $x^4$ equations?
5
votes
3answers
69 views

How can I prove that $f$ doesn't have all real roots $\forall a\in\mathbb{C}$

We have $f=x^4+ax^3+4x^2+1\in\mathbb{C}[x]$ with $x_1,x_2,x_3,x_4\in\mathbb{C}$. We need to prove that $\color\red{\forall a\in\mathbb{C}},f$ doesn't have all real roots. How can I begin to solve ...
3
votes
0answers
43 views

Fine the value of $P(n+1)$ given values of $P$ from 1 to $n$ [duplicate]

$P(x)$ is a polynomial of degree $n$ that satisfies $P(k)=\frac{k}{k+1}$ for $k=0,1,2,3,...,n$. Find $P(n+1)$. What have I tried: I have literally no idea how to do questions of this kind. Also, in ...
0
votes
1answer
17 views

About the solvability of certain equations

How one can see if this equation has real solutions: $$x^{2^{k}}-x-a=0$$ where $x$ is the unknown, $k$ is a positive integer and $a$ is a real constant.
3
votes
3answers
109 views
+50

A lot of confusion in the “Polynomial Remainder Theorem”?

Lately I've been reading about Polynomial Remainder Theorem from various sources, mainly from the wikipedea article, this post and some high school books. Wikipedea says that if we divide a polynomial ...
1
vote
1answer
50 views

Showing a polynomial is irreducible over $\mathbb{C}[x,y]$

Given $m,n \in \mathbb{N},$ how can I show that the polynomial $x^m+y^n-1$ is irreducible in $\mathbb C[x,y]$? I'm given the following hint, but I don't follow. Note: I know Eisenstein's ...
-1
votes
1answer
31 views

Finding polynomial when certain condition is given [closed]

When a polynomial 𝑃(𝑥) is divided by 𝑥^3 + 2𝑥^2 − 13𝑥 + 10, the remainder is 2𝑥 + 1; in addition, when 𝑃(𝑥) is divided by 𝑥^3 + 4𝑥^2 − 15𝑥 − 18, the remainder is 2𝑥^2 − 𝑥 − 4. Find the ...
0
votes
0answers
11 views

List of graph data points, need accurate calculation/estimate formula for excel

So I'm working on a small project that will use a calculation in excel, I'm nearly there and the only piece missing is being able to accurately estimate a data point from the following set of data ...
0
votes
2answers
48 views

Find $a$ and $b$ such that $f(x) = ax^3 + bx^2 - 28x + 15$ is divisible by $(x+3)$, etc

Find the values of $a$ and $b$ if the polynomial $f(x) = ax^3 + bx^2 - 28x + 15$ is exactly divisible by $(x+3)$ and leaves a remainder of $-60$ when $f(x)$ is divided by $(x-3)$. Use these values ...
2
votes
1answer
42 views

Irreducible polynomial modulo 2

I need to prove that polynomial $f(x) = x^{10}+x^{3}+1$ is irreducible modulo $2$. It is irreducible if $f|x^{1024}-x$, isn't it? I can use polynomial long division to check it, but this is not ...
2
votes
1answer
43 views

For which polynomials $P$ the integral $\int_0^\infty x^{z-1} P(x)^{-s} dx$ is computable?

I consider the following integral: $$ I(z,s)=\int_0^\infty \frac{x^{z-1}}{(P(x))^s}dx, $$ where $P(x) = a_0 + a_1 x + \cdots + a_n x^n$ is a polynomial of degree $n \geq 2$ with $P(x) > 0$ for ...