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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Working with casus irreducibilis

I read about casus irreducibilis here. As an example of casus irreducibilis, it says we can factor $x^3 - 15x - 4$ to find $4$ as a root and it also has two other real roots. Using Cardano's method we ...
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2answers
73 views

$x\in \mathbb{R} : P(x) \in \mathbb{Z} \Leftrightarrow Q(x) \in \mathbb{Z}$. Prove that $P(x)-Q(x)=c \in \mathbb{Z}$ or… [closed]

$P(x),Q(x)$ are two polynomials such that $x\in \mathbb{R} : P(x) \in \mathbb{Z} \Leftrightarrow Q(x) \in \mathbb{Z}$. Prove that $P(x)-Q(x)=c$ or $P(x)+Q(x)=d, $ where $c,d \in \mathbb{Z}$.
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why there are not polynomials $p,q$ such that $\sqrt{x^2-4}=\frac{p(x)}{q(x)}$

show that there are not polynomials $p,q$ such that $$\sqrt{x^2-4}=\dfrac{p(x)}{q(x)}$$ there a book say it is clear,because if such polynomials existed,then each zero of$x^2-4$ should have even ...
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1answer
344 views

Newton backward interpolation in Mathematica

I have the following task: Create a function (in Wolfram Mathematica), called $\mathrm{NewtonBackward}$[n_,x0_,h_,f_] which interpolates backwards the function $f(x)$ with nodes {x_i = x_0 + ...
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0answers
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An ideal in a ring of polynomials and a field extension.

Let $K\subseteq L$ be fields and $I$ an ideal of $K[x_1,...,x_n]$. I want to show that $IL[x_1,...,x_n]\cap K[x_1,...,x_n] =I$. The inclusion $I \subseteq IL[x_1,...,x_n]\cap K[x_1,...,x_n]$ is ...
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2answers
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Sufficient and essential condition for polynomials $P$ and $Q$ to satisfy $P(\sin x)= Q(\cos x)$

The famous identity $\sin^2 x+\cos^2x =1$ can be written as follows: The polynomials $P(x)=x^2$ and $Q(x)=1-x^2$ satisfy $$P(\sin x)= Q(\cos x),\quad \text{for all }x\in\mathbb R$$ What are ...
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2answers
57 views

Let a polyminal : $P(x)$ is a irreducible in $\mathbb{Q}[X]$. If $x_0 \in \mathbb{R} :P(x_0)=0$ prove that $P'(x_0) \not=0$ [closed]

Let a polyminal : $P(x)$ is a irreducible in $\mathbb{Q}[X]$. If $x_0 \in \mathbb{R} :P(x_0)=0$ prove that $P'(x_0) \not=0$ Vietnam 2014 (College)
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1answer
36 views

Inverse pairing function with polynomial constituents

Many bijective pairing functions $f:\mathbb N \times \mathbb N \rightarrow \mathbb N$ exists, including polynomial ones such as the Cantor pairing function $$f(n,m) = \frac{1}{2}(n + m)(n + m + ...
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1answer
105 views

GCD of a bivariate polynomial and its partial derivative..

I am stuck in the following question :- $f(x, y)$ is a bivariate polynomial with coefficients in $Z$. We have to show that $deg(GCD(f, f_y)) > 0$ iff $deg(GCD(f, f_x)) > 0$.(Here $f_x$ denotes ...
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2answers
41 views

$x^3-3x^2+(a^2+2)x-a^2$ has 3 roots $x_1,x_2,x_3$ such that $\sin \tfrac{2\pi x_1}{3}+\sin \tfrac{2\pi x_3}{3}=2\sin \tfrac{2\pi x_2}{3}$. Find $a$.

$x^3-3x^2+(a^2+2)x-a^2$ has 3 roots $x_1,x_2,x_3$ such that $\sin \dfrac{2\pi x_1}{3}+\sin \dfrac{2\pi x_3}{3}=2\sin \dfrac{2\pi x_2}{3}$. Find $a$ (Bulgari 1998)
2
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1answer
59 views

Roots of $z^5 (z − 2) = w $ in Unit disk

The Q is following : Prove that for each w in the unit disc $D(0, 1)$, the equation $z^5 (z − 2) = w $ has exactly five solutions in the unit disc counted with multiplicity. My Approach : let $f(z) ...
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1answer
29 views

Proving a linear transform defined by an integral is injective

Let the fact that $I(p)(x)=\int_0^x p(s) ds$ is a linear transform from $P_4\rightarrow P_5$ be given. Prove that $I$ is injective. Would it be sufficient to just state that for any 2 ...
2
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2answers
30 views

Existence of polynomials $g$, $h$, with non-negative coefficients, such that $f(x)= \frac{g(x)}{h(x)}$ [closed]

Suppose $a$ and $b$ are real numbers such that the quadratic polynomial $f(x) = x^2 + ax + b$has no non-negative real roots. Prove that ther exist two polynomials g,h, whose coefficients are ...
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The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate ...
3
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2answers
28 views

Elementary bound theorem of a monic real polynomial

An elementary bound theorem on the roots of a real monic polynomial states that $$M := \operatorname{max} (1, |a_0| + \cdots + |a_{n-1}|) := \operatorname{max} (1, B)$$ is an upper and lower ($-M$) ...
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0answers
90 views

Formula for finding the Roots of a cubic polynomial and nature of roots depending on the discreminant

I am trying to find the roots of a cubic polynomial in variable $r$: $ar^3 - r^2 +2mr -P^2=0$, $P$, $m$ and $a$ are constants here. I know that the discriminant of this polynomial for cubic roots is: ...
7
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1answer
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GCD of two polynomials in $ \mathbb{Z} [X]$

Let $(P,Q) \in ( \mathbb{Z} [X])^2$, such that $P$ and $Q$ don't have a common complex root, show that the sequence $\gcd(P(n),Q(n))_{n\ge0}$ is periodic. It seems to be a hard problem, please help. ...
2
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1answer
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If $P''(x)\mid P(x)$ then $P(x)$ has all roots real or less than $3$. [closed]

Let $P(x)$ be a polynomial of degree $n$ with real coefficients, such $P(x)$ has more than $3$ real roots. Assume that $P''(x)\mid P(x)$. Prove that $P(x)$ has $n$ real roots.
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Shamir's secret sharing interpolation problem

I try to understand this protocol - Shamir's secret sharing - threshold scheme. I got my data and I made interpolation basing on examples published on Wikipedia. You can see them below (sorry, I am ...
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2answers
70 views

Consider $n$ numbers $a_1,…, a_n$ and $x_1,…, x_n$. Can one find a polynomial, $f(x)\in R[x]$ st $f$ path through $(x_i,a_i) $

Consider $n$ arbitrary integer numbers $a_1,\ldots, a_n$ and real numbers $x_1,\ldots, x_n$. Can one find a polynomial, $f(x)\in \mathbb{R}[x]$ such that the graph of $f$ path through $(x_1,a_1), ...
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5answers
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Prove that equation $x^6+x^5-x^4-x^3+x^2+x-1=0$ has two real roots

Prove that equation $$x^6+x^5-x^4-x^3+x^2+x-1=0$$ has two real roots and $$x^6-x^5+x^4+x^3-x^2-x+1=0$$ has two real roots I think that: ...
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0answers
15 views

Finite Inseparable Extension

Preparing for my Galois theory exam in may and i am faced with the following question. Give an example of a finite inseparable extension with a sketched proof of its inseparability I have the ...
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3answers
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$P,Q$ are polynomials with real coefficients and for every real $x$ satisfy $P(P(P(x)))=Q(Q(Q(x)))$. Prove that $P=Q$

$P,Q$ are polynomials with real coefficients and for every real $x$ satisfy $P(P(P(x)))=Q(Q(Q(x)))$. Prove that $P=Q$. I see only that these polynomials are same degree
2
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2answers
38 views

$f(x)$ is a polynomial with complex co-effcients taking integer values for all sufficiently large integers $x$ , then $f$ integer for all integers?

Let $f(x)$ be a polynomial with complex co-effcients such that $\exists n_0 \in \mathbb Z^+$ such that $f(n) \in \mathbb Z , \forall n \ge n_0$ , then is it true that $f(n) \in \mathbb Z , \forall n ...
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1answer
36 views

relations between a set of polynomials

I have a set of polynomials. Is there a computer algebra program that gives all the algebraic relations between them ? I will prefer singular if it has this component.
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2answers
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Let $ (x-1)^n\mid P(x)$ Prove that $P(x)$ has $n+1$ coefficients not zero

Let $ (x-1)^n\mid P(x)$ Prove that $P(x)$ has $n+1$ coefficients not zero It's is 1977 Bulgaria contest, i tried but not succeed
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2answers
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Given some zeroes of a real polynomial of a given degree, how can one find the remaining zeroes?

Here is what the problem says: If $2$, $-\sqrt{5}$, and $3+i$ are three zeroes of a $5$th degree polynomial function with real coefficients, find the other zeroes of multiplicity $1$. I don't ...
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0answers
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Finding polynomial generators in a subspace

$S$ is a subspace $S= \{p\in P_3|~\text{$i\in\Bbb C$ is root of $p$}\}$. So the question at hand is how do you find the system of generators for the subspace knowing that $x$ is $p$'s divisor? ...
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Create a fourth order polynomial function f(x,y) with at least two distinct terms

I will be computing the gradient, finding the critical points, and use Lagrange multipliers to either maximize or minimize the function. Any suggestions?
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1answer
21 views

On the leading co-efficient of polynomial which takes integer values at every integer argument

If $f(x)$ is a polynomial with complex co-efficients of degree $k$ with leading co-efficient $a_k$ such that $f(n) \in \mathbb Z , \forall n \in \mathbb Z$ , then is it true that $|a_k| \ge \dfrac ...
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0answers
38 views

linear independence on polynomials

Suppose that $p_0,p_1,\dots,p_m$ are polynomials in $p_m(\Bbb F)$ such that $p_j(2)=0$ for each $j$. I want to prove that ($p_0,p_1,\dots,p_m)$ is not linearly independent in $P_m(\Bbb F)$ Now this ...
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1answer
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Quadratic graph / standard form

If I draw a graph of the quadratic $x^2-9=0$, I have a parabola with roots $x=3$ and $x=-3$ and a vertex of $(0,-9)$ with the parabola opening upwards as $a$ is positive in the standard quadratic ...
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0answers
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Taylor polynomial to find an approximation

Use the Taylor polynomial of degree 5 to give an approximation for ln(2) This may seem really simple but I have no idea how to do it, please help.
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Product of monic polynomials in finite fields

I am trying to show that the product of monic polynomials of degree $n$ in $\mathbb{F}_p[T]$ is given by $\prod_{i=0}^{n}(T^{p^n}-T^{p^i})$. I tried generating function but with no luck. Any hint?
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Show any straight line is irreducible

Show that any straight line in $\mathbb{F}^{n}$ is irreducible, where F is an infinite field. I know V($ax+b$) would be a variety that represents any straight line and then V is irreducible if I(V) ...
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zariski closure of first quadrant

Consider the boundary of the first quadrant in $\mathbb{R}^{2}$. Show that this is not a variety, and then find its Zariski closure. So as we are looking at the first quadrant we can write $$S= ...
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2answers
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Graphing: Given two points on a graph, find the logarithmic function that passes through both.

Is there such a method to do this? I would like to come up with a logarithmic function (a graph that looks like a square root graph) that passes through two given points. Haven't had any luck in ...
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1answer
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Determining whether two 2D polynomial curves are everywhere close to each other

Let's say we have two curves $P(t), Q(t): [0, 1] \to \mathbb{R}^2$. $P_x(t), P_y(t), Q_x(t), Q_y(t)$ are all polynomials of some degree $n$. We can further restrict this to Bernstein basis polynomials ...
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0answers
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Decomposable polynomails of 2 degree in finite space $\mathbb{F}[X]$

How can I show that there is a decomposable polynomial of second degree in a finite space $\mathbb{F}[X]$? I tried contrapositive proof but I got stuck. That made me think that maybe I should go for ...
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2answers
38 views

solving the equation where two variables are used

Solve the equation $$\frac{x}{x-a} + x = \frac{b}{b-a}+ b$$ The equation doesn't make sense. Should we take the LCM .only one equation and two variabkes are given
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1answer
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Can a quartic equation be reduced to a cubic/quadratic knowing that two roots are real?

I have a quartic equation that is the determinant of a 4-by-4 matrix that looks like: $det(M-\lambda I) = det \left( \matrix{m_{11}-\lambda & m_{12} & m_{13} & 0 \\ m_{21} & ...
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2answers
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Polynomials and partitions

There is a question I have based on the fact: If you take a quadratic polynomial with integer coefficients, take the set $\{1,2,3,4,5,6,7,8\}$, make a partition $A=\{1,4,6,7\}$, $B=\{2,3,5,8\}$, and ...
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Remainder from polynomial long division

Let $n>1$ and $n\mid p-1$ where $p$ - prime number. Prove that the remainder of dividing $x^p-x$ into $x^n-A$ is equal to $x(A^{\frac{p-1}{n}}-1)$ My sketch proof: We will write this in such form ...
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1answer
30 views

Finding normal basis for GF(q^m) over GF(q)

Could you kindly explain, how can one find a normal basis for GF$(3^6)$ over the GF$(3^2)$? As I understood, I should start with finding the polynomial in a form $$a(x^2) + (a^9)x + a^{81},$$ which ...
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2answers
37 views

Factor the polynomial $x^4 + 2x − 4$ in $\mathbb{Z}_5[x]$.

I'm confused as to how this is different from factoring in the reals? Would I start this by writing $x^4+2x-4 \equiv 0 \pmod 5$? What changes?
4
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1answer
93 views

Writing the roots of a polynomial with varying coefficients as continuous functions?

Consider the monic polynomial $$p_{\zeta}(z) = z^n + a_{n-1}(\zeta)z^{n-1} + \dots + a_0(\zeta), $$ where the $a_{i}$'s are continuous functions defined over $\mathbb{C}$. As is well known, the ...
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2answers
66 views

Solution of recursive polynomial functions

Is there anything that can be said about the roots of the polynomial $f_n(x)$ if $f_n(x) = xf_{n-1}(x) + f_{n-2}(x)$ where these are polynomials of degree $n, n-1,$ and $n-2$, respectively? My goal is ...
0
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4answers
66 views

How to prove that ${1,x,x^2}$ is a basis of a real polynomial functions space.

Let $V$ be the real vector space of all polynomial functions from $\mathbb{R}$ to $\mathbb{R}$ at most second degree. That is, the space of all functions with form $f(x)=c_0+c_1x+c_2x^2$ with ...
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1answer
26 views

Polynomial LongDivision

What would be the result of $x^3-4x^2-5$ divided by $x-3$ ? I am getting $4$ as my solution can someone prove me wrong, this is very confusing.
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0answers
25 views

How to upper-bound the smallest positive root of a polynomial?

Is there any algorithm for (upper-)bounding the smallest positive root of a polynomial of an arbitrary degree if it exists, or detecting that it does not exist otherwise? Edit: I'm looking for a ...