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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Polynomial factorization to irreducible factors with respect to field

I have a question, I think I don't understand this material very well and could use an explanation / some help. Basically we are asked to decompose $x^5-x$ to irreducible factors over $R,F2,F5,C$ ...
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70 views

Algebra 2 - Imaginary roots of Polynomials

Question: One zero of $P(z) = z^3 +az^2 + 3z + 9$ is purely imaginary. If $a \in \mathbb{R}$, find $a$ and hence factorize $P(z)$ into linear factors. What I've done: I know that the $P(z)$ is ...
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33 views

Compute all the directional derivatives of a trivariate polynomial function quickly

Given a trivariate polynomial $A\in\mathbb{R}[x,y,z]$, a direction $\vec v\in\mathbb{R}^3$ and a point $p\in \mathbb{R}^3$, what is the fastest way to compute the directional deriviatives ...
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polynomial division, gcd, question

We are asked to show that there are polynomials $p,q \in Q[t]$ such that: $p(t)*(t^4+2t^2+1)+q(t)*(t^4-3t^2-4) = t^2+1$ Is the answer the same for $t+5$ instead of $t^2+1$? What I tried doing: I ...
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58 views

Any finite set in $k^n$ is an algebraic set.

I'm trying to show that given a field $k$, and a finite set of points $\{a^i: i = 1\dots n\} \subset k^n$ is an algebraic set or equivalently is the set of common zeros of some set of polynomials $S ...
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39 views

How do we calculate the Euler numbers of this

Suppose we are given two cubics X(a) and Y(a) in $CP^2$; $X(a)={ (4-a^3) xyz-a^3(x^3+y^3+z^3) =0 }$ $Y(a)={ a(x^3+y^3+z^3)-(2+a^3)xyz =0 }$ where a is a parameter in C satisfying $a^3 \not=1$ and ...
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85 views

Minimize the mean square error $\frac{1}{m}\sum_{i=1}^m \| x_i - S(t_i)\|_2^2$ for a Bezier curve

The problem is 2.1 from here. I am trying to minimize the mean error $$ E(\alpha_1,\alpha_2) = \frac{1}{m}\sum_{i=1}^m \| x_i - S(t_i)\|_2^2 $$ Where $x_i$, lie on the curve $\gamma$ and $S(t_i)$ ...
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Cubic Poynomial : In the equation $x^3 +3Hx +G=0$ if G and H are real and $G^2 +4H^3 >0$ then roots of the…

Question: In the equation $x^3 +3Hx +G=0$ if G and H are real and $G^2 +4H^3 >0$ then roots of the equation are (a) all real and equal (b) all real and distinct (c) one real and two ...
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How to write product of three sums

I know that by the binomial theorem, $\displaystyle \left(\sum_{n=0}^\infty a_nx^n \right)\left(\sum_{n=0}^\infty b_nx^n \right)= \sum_{n=0}^\infty \left(\sum_{k=0}^n a_kb_{n-k} x^n\right)$. How do I ...
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Express $cos2\theta$ in terms of $cos$ and $sin$ (De Moivre's Theorem)

Use De Moivre's to express $cos2\theta$ in terms of powers of $sin$ and $cos$ What I have is: $cos2\theta + isin2\theta\\ = (cos\theta + i sin\theta)^2\\ = cos^2\theta + 2 cos\theta i sin\theta + (i ...
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60 views

Proving that a polynomial about the volume of a tetrahedron is irreducible

We know that the volume of a tetrahedron $ABCD$ can be represented as ...
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57 views

interchanging the variables in the equation

I am given the following equation $$y=7.515x^3-10.229x^2+5.05x$$ How do I find the value of $x$ when $y$ is given. I need to somehow interchange the position of $x$ and $y$ so that the ...
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Rewriting algebra

I'm working on calculations on polynomials and a paper gives the following algebra step: $\frac{x^{15} - 1}{x^3 - 1} = x^{12} + x^9 + x^6 + x^3 + 1$ They do not explain how they get to this result, ...
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226 views

Show that a polynomial has at least one positive solution/root.

Let: $P(x) = a_nx^n + a_{n-1}x^{n-1} + ..... + a_1x + a_0$ where $a_0a_n < 0$ I have to prove that the Polynomial $P(x)$ has at least one positive root how can I prove it? Any ideas?
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Dimension of local ring as vector space over $\mathbb C$

I want to know what the dimension of each of the local ring $\mathbb C[x,y]_p/(y^2-x^7,y^5-x^3)$ is, where $p\neq (0,0)$ over $\mathbb C$-vector space. I know the dimension of it in the origin point, ...
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74 views

How to prove that this polynomial has no more than $s$ repeated roots

Let $\beta_{1},\beta_{2},\cdots,\beta_{s+1}\in R$,and $\alpha_{0},\alpha_{1},\cdots,\alpha_{s}$ be postive integers, with $\alpha_{0}>\alpha_{1}>\cdots>\alpha_{s}$. Show that: the ...
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$\frac{x^4 - x^3 + ax^2 + bx + c}{x^3 + 2x^2 - 3x + 1}$, remainder $3x^2 - 2x + 1$. Find $(a + b)c$.

Given the polynomials $P(x) = x^4 - x^3 + ax^2 + bx + c\\ Q(x) = x^3 + 2x^2 - 3x + 1\\ R(x) = 3x^2 - 2x + 1$ such that $P(x) = D(x)Q(x) + R(x)$, find $(a + b)c$. I would normally apply little ...
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homework - Show a matrix as a combination of other matrices and long division

The topic we are dealing with here is polynomial division. The question is: We are given a polynomial: $f(x) = (x+1)(x-1)^2$, and a matrix $D \in R^{nxn}$ such that $f(D)=0$ Using only $I, D, D^2$ ...
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Galois group of $x^5-2px+p$ over $\mathbb{Q}$ with $p$ prime

I proved that $x^5-2x+p$ is irreducible in $\mathbb{Z}$ so in $\mathbb{Q}$ by Gauss Lemma I need (and I still can't) to prove that $x^5-2x+p$ has exactly three real roots and conclude that the Galois ...
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Given $A, B\in R^{n\times n}$ diagonal matrices, there exist $p,q \in R[x]$ and $X\in R^{n\times n}$ such that $A = p(X),B=q(X)$

(1) We are given $A,B \in R^{n\times n}$ diagonal matrices of n rows and n columns with real values. Show that there are $X \in R^{n\times n}$ and polynomials $q$ and $p$ such that: ...
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80 views

Identity between roots of polynomials

Let $A\in{\mathbb C}[X]$ be a monic polynomial of degree $n\geq 2$, with roots $\alpha_1,\alpha_2,\alpha_3, \ldots ,\alpha_n$. Let $B$ be the polynomial $$ B=\prod_{k=1}^{n} ...
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The kernel and image space of $D$ and defining $D$ as a matrix product equation.

There is an isomorphism between $P_n(x) = \{p(x) : p(x) = a_0 + a_1x + a_2x^2 +\ldots+ a_nx^n,\ \forall a_i \in \Bbb R\}$ and $\Bbb R^{n+1}$, in the sense that $a_0 + a_1x + a_2x^2 + ... ...
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1answer
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given the polynomial function find the function values

I given a problem and I can not get to the answer that I was given by my professor. P(x) = -x^2 - x + 12 where P(-2). I keep coming up with 10. He gave me the answer of -2. Am I missing something?
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Finding roots of a function in an interval

Does the equation $x^3-12x+2=0$ have three solutions in the interval $[-4,4]$? We know that this is a continuous function because it's a polynomial, and so we can use the Intermediate Value ...
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Fields of polynomials . Proving that a belongs to k as a root

if $f(x)\in k[x]$, where $k$ is a field, then $a\in k$ is a root of $f(x)$ iff $x-a$ divides $f(x)$ in $k[x]$. My result ... If $a$ is a root of $f(x)=q(x)(x-q)$ and if we let $f(x)=q(x)(x-a)$,then ...
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1answer
140 views

Krull dimension of this local ring

I want to know what the Krull dimension of this ring $\mathbb C[x,y]_p/(y^2-x^7,y^5-x^3)$ is, where $p\neq (0,0)$. I know the dimension of it in the origin point, but I don't know other cases.
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Finding $a_n$ such that $x^n+a_1x^{n-1}+\cdots+a_{n-1}+a_n$ cannot be factored when $a_1,\cdots,a_{n-1}$ given

Let $n\ge 4\in\mathbb N$. Suppose that $a_1,a_2,\cdots,a_{n-1}$ are given integers. Then, here is my question. Question : Is the following true for any $(a_1,a_2,\cdots,a_{n-1})$ ? There ...
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1answer
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How rapidly can a polynomial grow in a proximity of the real segment comparing to the values on the segment?

Let $P_n$ be a polynomial of degree $n$ with complex coefficients. Does for any $l>0$ and small $\varepsilon>0$ there exist $C=C(l,\varepsilon)>0$ and $q=q(l,\varepsilon)>1$ s.t. in the ...
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Polynomial roots of degree 2

Since every polynomial of degree 'n' has 'n' complex roots. Then what about $P(x)=x^2$ . Isn't $x=0$ the only possible root of this polynomial ?
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Extended GCD of polynomials

Say $H(x),F(x),G(x)\in\Bbb Q[x]$ with degrees $h$, $f$ and $g$ respectively. Let: $1)$ $f,g<h<f+g$ $2)$ $gcd(F(x),G(x))=1$. $$\mbox{Extended GCD }\implies\exists A(x),B(x)\in\Bbb ...
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Is there a named theorem for the fact that two points define a line, and three points define a quadratic function?

In particular, is there a theorem stating the fact that a polynomial function of degree d is defined by d+1 points? I'm asking ...
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Find a polynomial $p$ of degree $3$ if its value in $4$ points is given

Find a polynomial $p$ of degree $3$ such that \begin{align*} p(−4) &= −142, \\ p(1) &= −2, \\ p(−5) &= −242, \\ p(4) &= 10. \end{align*} Then use your polynomial to approximate ...
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1answer
430 views

Is my simple (in my opinion) way of solving cubic equations correct?

I've been analyzing ways of solving cubic equations and I've come up with this one. I've tried to make it as simple as possible. So I'll show you a way of solving cubic equations when none of the ...
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Defining irreducible polynomials recursively: how far can we go?

Fix $n\in\mathbb N$ and a starting polynomial $p_n=a_0+a_1x+\dots+a_nx^n$ with $a_k\in\mathbb Z\ \forall k$ and $a_n\ne0$. Define $p_{n+1},p_{n+2},\dots$ recursively by $p_r = p_{r-1}+a_rx^r$ such ...
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Interpolation- Barycentric coefficients for nodes that are Chebyshev points of the second kind.

So I came across the following theorem: If the interpolation node are Chebyshev points of the second kind given by : $$ x_k=\cos \left( \frac{j\pi}{n}\right) \qquad ( 0 \leq j \leq 0) $$ Then the ...
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Polynomials: irreducibility $\iff$ no zeros in F.

Given is the polynomial $f \in F[x]$, with $deg(f)=3$. I have to prove, that f is irreducible iff $f$ has no zeros in $F$. "$\Rightarrow$": let's prove the contrapositive: "if $f$ has zeros in $F$, ...
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The Limitations of Vieta's Formula

I was attempting to find the roots of $f(x)=2x^3+10x^2+5x−12$ and since the the OP had already found one of the roots, I tried to recall a relation to help me find the other two easily. The first one ...
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What polynomial transformation is this and how is it related to the original polynomial?

Let $f(X) = a_n X^n + \dots+a_0$. Then the Laplace transform of $f$ is $g(s) = \mathscr{L}\{f\}(s) = \frac{n! a_n}{s^{n+1}} + \dots + \frac{a_0}{s}$. If you now define the polynomial transform of ...
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1answer
140 views

Reducibility of Equivalent Polynomials

This feels like a trivial question but somehow I couldn't come up with an immediate solution. Given polynomial $P$ in a multivariate polynomial ring over some base field $F$, if $P$ is irreducible ...
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homework: rings, matrices and polynomials

$A,B$ are both $n \times n$ and diagonal matrices. Prove that there is a matrix $X$ which is $n \times n$, and polynomials $p$ and $q$ such that $A= p(X), B= q(X)$ Is this true for ANY 2 matrices (we ...
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1answer
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Irreducible polynomials: one of them must be of degree zero

I have the following definition of irreducible polynomials, given by my professor: If $p = f * g$ , where $f, g \in F[x]$ then $degree(f) = 0 \ \ or \ \ degree(g) = 0$ And then the following ...
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1answer
56 views

$p$-polynomial of $n$'th degree, $q(x)=p[x,x_1,x_2,…,x_k]$, prove that q has the same leading coefficient.

So I have a polynomial $p$ of $n$'th degree and q given by $q(x)=p[x,x_1,x_2,...,x_k]$, meaning that for $x$ it gives back the leading coefficient in interpolation of $p$ on points $x,x_1,...,x_k$. ...
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62 views

A physics related question about an infinitely long pipe.

This is a really nice question I found some days ago, so I translated it into English to share. Suppose we have a water pipe which is infinitely long, with water flowing in it. We know that if a ...
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3answers
344 views

angle between polynomials

let $v$ be the space of polynomials less than or equal to three and let $$\langle p,q\rangle = p(0)q(0)+p'(0)q'(0)+p(1)q(1)+p'(1)q'(1)$$ What is the angle between the polynomials $2x^3-3x^2$ ...
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1answer
127 views

Minimal Polynomial of $\alpha^2$

Having already proved that $p(x)=x^5 + x^2 + 1$ is primitive in $GF(2)$ and assuming that $\alpha$ is a primitive element representing a root of $p(x)$, I am trying to minimal polynomial of $\alpha^2$ ...
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Generalizations of the quadratic formula [duplicate]

The quadratic formula can be used to find the roots of any quadratic polynomial of the form $ax^2 + bx + c$: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ The derivation is simple enough and uses a ...
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2answers
117 views

The largest root of $-3x^3+24x^2+6x-9=0$

Since the polynomial has three irrational roots, I don't know how to solve the equation with familiar ways to solve the similar question. Could anyone answer the question?
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The Conjugate Roots Theorem for Irrational Roots

The Conjugate Roots Theorem for Irrational Roots states that for a polynomial $f(x)$ with integer coefficients, if a root of the equation $f(x) = 0$ is expressed as $a+\alpha$, where $a\in\mathbb{Q}$ ...
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478 views

Third degree polynomial with integer coefficient and three irrational roots

There are some polynomial with the above characteristic, and real roots of such polynomials cannot be found using rational number theorem and irrational conjugate theorem. The example of such function ...
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1answer
193 views

Roots of some modified Bernoulli polynomials

Update The polynomials are generated as follows: Where $B_n(x) = \sum_{k=0}^n {n \choose k} b_{n-k} x^k$ is used to generate standard Bernoulli polynomials, top plot is generated as follows: ...