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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48 views

Galois group, algebraic closure over maximal extension

Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. Let $\alpha \in \overline{\mathbb{Q}}\setminus \mathbb{Q}$ and let $K \subset \overline{\mathbb{Q}}$ be a maximal extension of ...
0
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0answers
29 views

Polynomial with certain property

Does there exist a polynomial $P$ over the rationals s.t. for every $k=1,2,\dots,2015$ the equation $P(x)=k$ has exactly $k$ different rational roots, dropping multiplicities of these?
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2answers
41 views

Find Maximum and Minimum value by two polynomial equations

Suppose there are $7$ real numbers say $A,B,C,D,E,F,G$ All we need to find the minimum and maximum value of $G$ satisfying the following two equations :- Sum of Numbers :- $A + B + C + D + E + F ...
3
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1answer
37 views

What is the remainder when polynomial $f(x)$ is divided by $(x+1)(x-3)$ when $f(-1) = -4$ and $f(3) = 2$?

A polynomial $f(x)$ gives remainder $2$ when divided by $(x-3)$ and gives a remainder $-4$ when divided by $(x+1)$. What is the remainder when $f(x)$ is divided by $(x^2 - 2x - 3)$? I have shortened ...
7
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2answers
112 views

Does this sequence of polynomials have a limit?

Consider the sequence of polynomials $p_n$ defined as follows: $p_n$ is the unique polynomial of degree $2n+1$ satisfying $$p_n(0) = 0$$ $$p_n(1) = 1$$ $$p_n^{(k)}(0) = p_n^{(k)}(1) = 0 \text{ for ...
1
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0answers
37 views

Methods for evaluating polynomials quickly

I am wondering what methods exist for effectively evaluating polynomials (manually or in the head) in a quick, efficient fashion. For example, one of my favorite methods is the "nested form of a ...
2
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3answers
48 views

Does there exist an explicit formula for the coefficient of $x^k$ in the square of a polynomials?

So let's say we have a polynomial $P(x)$ of degree $n$, and we have: $$P(x)=\sum_{k=0}^{n}a_k x^k$$ I know that if you square $P(x)$, you get: $$P(x)^2=\sum_{k=0}^n \sum_{l=0}^na_ka_lx^{k+l}$$ ...
0
votes
1answer
52 views

Highest common factors of polynomials

Let h be a hcf of $f, g \in K[x]$ Then there exists polynomials a and b such that $h = af + bg$ Can anyone explain this theorem to me intuitively?
3
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0answers
105 views

Simplification of recursive polynomials

Suppose I have known polynomials $p_i, i=0\ldots k-1$. I have the following horrific looking recursive equation: ...
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0answers
25 views

Let $P$ with real coefficients satisfies $|P(i)|<1$. Prove that there is a root $z=a+bi$ of $P$ such that $(a^2 + b^2 + 1)^2 < 4b^2 + 1$

A monic polynomial $P$ with real coefficients satisfies $|P(i)|<1$. Prove that there is a root $z=a+bi$ of $P$ such that $(a^2 + b^2 + 1)^2 < 4b^2 + 1$ One solution is: Let us write $P(x) = ...
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0answers
22 views

Chebyshev polynomials approximation - Is there a way to generalize this

In an exam I was given this question: let $f(x)=x^3$. We want to find the best linear approximation (best in the sense that the maximal error is minimized) of $f$ in the interval $[-1,1]$ using ...
5
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2answers
103 views

How to reduce a quartic form to a quadratic form with equal roots

Given a polynomial in $n$ variables of the form $$P(x_1,x_2,\dots,x_n)=\left(\sum_{i,j}a_{ij}x_ix_j+\sum_{i}b_{i}x_i+c\right)^2$$ is there a way to find a polynomial also in $n$ variables of degree ...
4
votes
1answer
118 views

Determining the Number of Zeros of a (Holomorphic) Polynomial $f:\mathbb{C}\to\mathbb{C} $ in each Quadrant.

Suppose $f(z)=z^4+2z^3+3z^2-z+2$. I would like to be able to determine the number of zeros (without using a CAS) $f$ has in each quadrant. I recently learned about the Argument Principle and Rouche's ...
2
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1answer
40 views

Question about polynomial quotient ring F[x]/(ax-b)

Let $F$ be a field. Fix elements $a \in F^{\times}$ and $b \in F$. Prove that the quotient ring $F[X]/(aX − b)$ is isomorphic to F. So, I am thinking to use the first isomorphism theorem, and try to ...
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0answers
27 views

Prove that the maximum in absolute value of any monic real polynomial of n-th degree on [-1, 1] is not less than $\frac{1}{2^{n-1}}$

One solution is: Note that equality holds for a multiple of the n-th Chebyshev polynomial $T_{n}(X)$ The leading coefficient of $T_{n}$ equals $2^{n-1}$, so $C_{n}(X) = \frac{1}{2^{n-1}}T_{n}(X)$ is ...
1
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1answer
44 views

Eisenstein's criterion and polynomials of degree zero

Is Eisenstein's criterion applicable for polynomials in $\mathbb Z[x]$ of degree zero? If $a \in \mathbb Z$, then it is irreducible for $a \ne \pm 1$, but then surely I can find a prime number ...
0
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0answers
26 views

Understanding Abel's Multiplicative Product:

I was reading: http://www.math.caltech.edu/~jimlb/abel.pdf And didn't understand why the first theorem stated on page 8 must be true. The problem is posed as follows: Define $q \in \Bbb{N}, p \in { ...
0
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1answer
10 views

Lagrange interpolating polynomial with n=4

I don't know if I got that one right and I can't find on the internet the correct expression for the Lagrange interpolating polynomial when n=4, using these expressions: Note: I'm using $x_1$ ...
4
votes
2answers
58 views

Polynomial Diophantine Equation

If $x$,$y$ $\in \mathbb Z$, find all the solutions of $$y^3=x^3+8x^2-6x+8$$ I have tried factorizing the equation but the polynomial on $\text{R.H.S.}$ doesn't have any integral roots. ...
0
votes
3answers
36 views

Decompose a polynomial: find $f(x)$ such that $h(x) = f(g(x))$

I try to make an algorithm that decomposes a polynomial, ie find $f(x)$ such that $h(x) = f(g(x))$ by knowing $h$ and $g$. For example, having : $h(x) = 112x^6 + 1232x^5 + 2772x^4 - 3388x^3 + 847x^2 ...
3
votes
1answer
61 views

irreducibility in $\mathbb Q[X]$

Are these polynomials irreducible in $\mathbb Q[X]$: 1) $x^4+3x^3+x^2-2x+1$ What I did: I reduced it modulo $3$, then saw that it has no roots, so then I checked all $9$ monic polynomials of ...
0
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1answer
46 views

An exercise about field automorphisms and ideals.

Consider a field $K$ and the $K$-algebra $K[x_1,\ldots,x_n]$ of polynomials in $n$ variables; $\mathfrak a$ is an ideal of $K[x_1,\ldots,x_n]$ and suppose that there exists a field $L\subseteq K$ ...
2
votes
1answer
24 views

Basis of $k[X_1,\ldots,X_g]$

How many monomios exist in $k[X_1,\ldots,X_g]$ of degree d? following my calculations is $g^d$, am I right? is this a monomial basis of the polynomial ring $k[X_1,\ldots,X_g]$ of degree $d$? Thanks ...
1
vote
1answer
34 views

Roots of polynomial equation $2a x^\gamma + ax^{\gamma - 1} - 2 = 0$

I would like to find roots of the following polynomial equation $$2a x^\gamma + ax^{\gamma - 1} - 2 = 0$$ where $a,\gamma>0$ (we might also assume that $\gamma \in\mathbb{N}$ if needed). Playing a ...
0
votes
1answer
43 views

Evaluate derivative of Lagrange polynomials at construction points

Assume, that we have points $x_i$ with $i=1,...,N+1$. We construct the Lagrange basis polynomials as \begin{align} L_j(x) = \prod_{k\not = j} \frac{x-x_k}{x_j-x_k} \end{align} Now according to my ...
3
votes
5answers
137 views

Why does $p(a)=0$ imply $(x-a) \mid p$?

There's something I've never understood about polynomials. Suppose $p(x) \in \mathbb{R}[x]$ is a real polynomial. Then obviously, $$(x-a) \mid p(x)\, \longrightarrow\, p(a) = 0.$$ The converse of ...
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0answers
27 views

prove existence and uniqueness of the partial fraction decomposition of the polynomial ring over a field

I have to prove that for $f,g\in F[X]$, $F$ a field, where $g$ is monic with prime factorization $g=g_1^{v_1}...g_n^{v_n}$ (where the $g_i$'s aren't pairwise associated prime elements), there is an ...
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0answers
15 views

Possible Relations between Properties of a Polynomial and its Periodic Points

Question: Let $f(x)$ be a polynomial in $\mathbb{Z[x]}$. Is there a relation between the property $P_i$ of $f$ and the number of its periodic points with period $p$ (x is a $p$-periodic point of $f$ ...
3
votes
2answers
68 views

Do all polynomials of degree n with indeterminate coefficients have Galois groups that are isomorphic to Sn?

I just finished reading "A Book of Abstract Algebra" by Charles C. Pinter and, as someone who is studying this independently, I was having some understanding issues and many questions. 1) Does every ...
4
votes
2answers
55 views

Math Contest Question with Polynomials

Prove that there does not exist a polynomial f(x) with integer coefficients for which f(2008) = 0 and f(2010) = 1867. This is a question from CMOQR (Qualifier for Canadian Math Olympiad , not the ...
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2answers
33 views

Determinant of Polynomial

I was reading some paper and it says 'Let $\Delta$ denote the determinant of the polynomials $P,Q$ and $R$ with respect to the basis $1,X,X^2$' ($P,Q$ and $R$ are degree 2 polynomials). And then I ...
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3answers
38 views

Find a polynomial given the remainders of the division of that polynomial with 3 other polynomials

A polynomial from $ \mathbb{C}[x]$ divided by $ x - 1$, $x + 1$, $ x -2$ has the remainders 2, 6 and 3. Find the remainder of the division of that polynomial by $(x-1)(x+1)(x-2)$ The degree of ...
0
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1answer
47 views

How do I prove the following theorem?

If $F$ is a field and $f(x) \in F[x]$, then $f(x)$ has no repeated roots if and only if $(f, f') = 1$, where $f'$ denotes the derivative of $f$.
3
votes
1answer
86 views

Geometry: How to find cube root, fourth root, fifth root… and so on?

As we know that square root of a number $n$ can be found by using a compass and a straight edge, given the line of length $n$. What I want to know is how to find cube root, fourth root, fifth root or ...
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votes
6answers
5k views

Proof the Fibonacci numbers are not a polynomial.

I was asked a while ago to prove there is no polynomial $P$ in $\mathbb R$ such that $P(i)=f_i$ for all $i\geq0$. I tried to get a proof as slick as possible and here's what I got. Let ...
6
votes
1answer
39 views

$f=X^3+aX+b$ has pairwise different roots iff $-4a^3-27b^2\neq0$

I have to proof that $f=X^3+aX+b\in F$, where $f$ is a polynomial which is a product of linear terms in a field $F$, has pairwise different roots iff $d=-4a^3-27b^2\neq0$. Now what I have done is ...
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2answers
41 views

Looking for the name of polynomials obtained as integrals over a simplex

I'm looking for the name of the following polynomials: $\mathrm{p}_1 = 1$ $\mathrm{p}_2 = x - \frac{1}{2}$ $\mathrm{p}_3 = \frac{1}{2} x^{2} - \frac{1}{2}x +\frac{1}{6}$ $\mathrm{p}_4 = \frac{1}{6} ...
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1answer
23 views

How to follow $n=m$ from $(t-a)^n p(t)=(t-a)^m q(t)$?

Say, we have $$(t-a)^n p(t)=(t-a)^m q(t)$$ where $p, q$ are polynomials and $p(a)\neq 0 \neq q(a)$. How can we conclude that $n=m$? I've tried: the setup implies that $$n + \deg p =m+\deg q$$ but how ...
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0answers
5 views

Bivariate polynomials at bounded evens

Given $n$, is there a total degree $cn^{c'}$ polynomial $p(x,y)\in\Bbb R[x,y]$ and a total degree $dn^{d'}$ polynomial $q(x,y)\in\Bbb R[x,y]$ with fixed $c,c',d,d'>0$ such that $$x,y\in\Bbb ...
2
votes
3answers
79 views

Proof that linear polynomial has exactly one root

How can one prove that all linear polynomials have exactly one root? This is geometrically intuitive (just rotate a line around the x axis) however I'm not sure how to formally prove this.
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0answers
39 views

Maximize area of rectangle fenced on three sides [duplicate]

A city decides to make a park by fencing off a section of riverfront property. Funds are allotted to provide $80$ meters of fence. The area enclosed will be a rectangle, but only $3$ sides will be ...
2
votes
1answer
61 views

Derivation of dice-sides polynomial

This question on PPCG.SE fascinated me. In particular @xnor's brilliant answer providing an elegant polynomial to provide the solution. In summary, the question asks for a program that, for an ...
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1answer
45 views

A question on elementary symmetric polynomials

Is it possible to express $$(a-b)^3+(b-c)^3+(c-a)^3$$ as a combination of elementary symmetric functions $a+b+c, ab+ac+bc$, and $abc$? Thanks a lot.
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2answers
31 views

Find a polynomial of degree 3 with rational coefficients, given remainders mod two quadratics [closed]

Find a polynomial of degree 3 with rational coefficients, which divided by $X^2 - 5x + 6$ has the remainder $2x - 1$, and at the division with $X^2 + 1$ has the remainder $x - 2$.
4
votes
1answer
45 views

Does there exist a polynomial over integers possesing certain property?

Does there exist a polynomial with integer coefficients which posseses the local minimal value $\sqrt{2}$ (not a local minimum at $\sqrt{2}$)?
0
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1answer
43 views

How to plot this complex polynomial?

I have tried using ti-89, ti-nspire and wolfram alpha, but none have given me a plot of this complex polynomial. I admit that I have may not inputted the formula correctly in wolfram alpha. The ...
0
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0answers
38 views

Equation with unknown with high pow value.

Please someone help me to isolate $b$ from the rest of equation where $d$ and $n$ are known values. $d$ and $n$ can be different in different cases. $$ nb - b^{d} = n + 1 $$
2
votes
1answer
71 views

How many roots does $(1+z^3)^8 = (1 + z^4)^6$ have?

I got stuck on this question: How many roots does $(1+z^3)^8 = (1 + z^4)^6$ have? (including complex roots and roots with multiplicity) My attempt at a solution: First we can write the equation ...
0
votes
0answers
18 views

Finding a homeomorphism between quadratic polynomials

I would like to represent a quadratic polynomial $f(x)=ax^2+bx+c$ as $$f=\phi\circ f_\lambda \circ \phi^{-1},$$ where $f_\lambda(x)=\lambda x(1-x)$ with $\lambda = 1+\sqrt{(b-1)^2-4ac}$. Is this ...
0
votes
4answers
32 views

Polynomial Function Equation

I am having some trouble with a problem. Thanks in advance to anyone who answers. Find a and b if $4x^4 + ax^3 + bx^2 + 6x + 1 = |P(x)|^2$ I have been staring at this problem for quite some time, ...