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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Factorization with a Primitive Factor of Polynomials

Question: Let $f,g\in\Bbb Q[x]$. Why is it that $\rm\color{#c00}{(1)}$ if $f$ is monic then $f=\frac{1}{a}f^*$ for some primitive polynomial $f^*\in\Bbb Z[x]$ and $a\in\Bbb Z$ ? ...
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433 views
+100

Finding the all roots of a polynomial by using Newton-Raphson method.

Is there a general formulation for finding all roots of a polynomial, especially the complex ones, by using the Newton-Raphson Method?
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0answers
43 views

Real roots of an nth order polynomial

Given an nth order polynomial, is there any algorithm that can calculate all the roots ? Is there any algorithm that can calculate ALL the roots of the equation ? ...
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3answers
41 views

Linear Algebra: Polynomials Basis

Consider the polynomials $$p_1(x) = 1 - x^2,\;p_2(x) = x(1-x),\;p_3(x) = x(1+x)$$ Show that $\{p_1(x),\,p_2(x),\,p_3(x)\}$ is a basis for $\Bbb P^2$. My question is how do you even go about proving ...
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50 views

Limits using Maclaurins expansion for $\lim_{x\rightarrow 0}\frac{e^{x^2}-\ln(1+x^2)-1}{\cos2x+2x\sin x-1}$

$$\lim_{x\rightarrow 0}\frac{e^{x^2}-\ln(1+x^2)-1}{\cos2x+2x\sin x-1}$$ Using Maclaurin's expansion for the numerator gives: $$\left(1+x^2\cdots\right)-\left(x^2-\frac{x^4}{2}\cdots\right)-1$$ And ...
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2answers
37 views

Getting the multiplicative inverse of a polynomial

I have a polynomial $m(x)= x^2 + x + 2$ that's irreducible over $F=\mathbb{Z}/3\mathbb{Z}$. I need to calculate the multiplicative inverse of the polynomial $2x+1$ in $F/(m(x))$. I'd normally use ...
6
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1answer
67 views

When does a formula for the roots of a polynomial exist?

My question is straightforward to pose: given a polynomial $f$ over a subfield of $\mathbb{C}$, are there conditions which guarantee the existence of a closed formula for the roots of $f$ in terms of ...
6
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0answers
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A polynomial that annihilates two other

While studying, I found the following problem: Let $f, g \in F[t]$. Prove that $\exists p \in F[x, y], p \neq 0 : p(f(t), g(t)) = 0$ I'd thank any hints that point me in the right direction.
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2answers
31 views

how find this polynomial? [on hold]

Determine the polynomial $f(x) \in \mathbb{R}[X]$ with degree 3 and satisfies the following conditions: $f(0)=0$ and $f(x-1)=f(x)+(2x)^{2}, \forall x \in \mathbb{R}$.
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2answers
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Mclaurins with $e^{\sin(x)}$

To evaluate $e^{\sin(x)}$ I use the standard series $e^t$ and $\sin(t)$, combining them gives me: $e^t = 1+t+\dfrac{t^2}{2!}+\dfrac{t^3}{3!}+\dfrac{t^4}{4!}+O(t^5)$ $\sin(t) = ...
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1answer
44 views

Irreducible in $\Bbb Q[x]$

Suppose $f(x)$ is an polynomial of integer coefficients. If for infinitely many integers $x$, $f(x)$ is prime. Show that $f(x)$ is irreducible in $\Bbb Q[x]$. Suppose $f(x)$ is is reducible in $\Bbb ...
4
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2answers
181 views

Proof polynomial has only one real root.

I need to prove that this polynomial equation: $$x^5-(3-a)x^4+(3-2a)x^3-ax^2+2ax-a=0\quad\text{ for }\quad a\in(0,\frac{1}{2}).$$ has only one root. That it has one real root is obvious because it is ...
2
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0answers
27 views

Objects corresponding to Higher forms

If $Q$ is a quadratic form, then we know there exists matrix $A$ such that $Q=xAx'$ and $Q$ can be expressed as weighted sum of eigenvalues of $A$. If $H$ is a higher order form, then is there an ...
2
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2answers
16 views

Polynomials and permanent/determinant

Let $f\in \Bbb Z[x_1,\dots,x_n]$ be a multivariate polynomial. Is it possible to represent $f$ say of TOTAL degree $d$ by a $({dc})^{n}\times ({dc})^{n}$ determinant or $({dn})^c\times ({dn})^c$ ...
2
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2answers
28 views

Question about diagonalization and projections

Let a finite dimensional vector space $V$ above $\mathbb{F}$. Let $T:V\to V$ a diagonlizable transformation. We denote $a_1 \ldots a_r$ the $r$ different eigenvalues of $T$. By diagonalization, we ...
2
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1answer
24 views

Polynomial functions/basis

If I suppose $R \subset F$ and have polynomial functions $p_{k,j} : F \to F$ by $p_{1,0}(x)=(x-2)^3$ $p_{2,0}(x)=(x-1)$ $p_{2,1}(x)=(x-1)(x-2)$ $p_{2,2}(x)=(x-1)(x-2)^2$ and the polynomial ...
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3answers
116 views

Using Rouche's Theorem to find the number of zeros of $p(z)=z^8 +10z^3 −50z+1$ in the right halfplane

I'm studying for a complex analysis qualifying exam and was wondering if someone could help me out with this. I am not sure how to apply Rouche's Theorem to this. How many zeros does the polynomial ...
5
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0answers
34 views

Irreducibility of $~\frac{x^{6k+2}-x+1}{x^2-x+1}~$ over $\mathbb Q[x]$

The Artin—Schreier polynomial $~x^n-x+1~$ is always irreducible over $\mathbb Q[x]$, unless $n=6k+2$, in which case it seems to have only two factors, one of which is always $x^2-x+1$. The ...
3
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1answer
51 views

Polynomial Pell equation

Can someone point me in the right direction? Let $k$ be a field of characteristic $0$ and let $D \in k[x]$ be non-constant. Prove that the ‘polynomial Pell’ equation $$f^2 − Dg^2 = 1,\,\,\,\,f, ...
1
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1answer
42 views

Is $x^3$ in the null space of the transformation $p(x) \mapsto xp(x)$?

Let $h: P_3 \to P_4$ be given by $p(x) \mapsto xp(x)$. Is $x^3$ in the null space ? Or is it in the range space ? Also, I am having difficulty finding the null space and the range of this map, can ...
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2answers
53 views

Prove particular quintic is irreducible

The problem is to prove that the quintic $$x^5+10x^4+15x^3+15x^2-10x+1$$ is irreducible in the rationals. I don't have much knowledge in group theory, and certainly not in Galois theory, and I'm ...
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348 views

Is there a rule to the terms of a falling factorial?

$\require{cancel}$I discovered that $n!=\xcancel{(n)_{n-1}}n^{\underline{n-1}}=n(n-1)(n-2)\cdots(3)(2)$. I have expanded a few examples: $$2!=\xcancel{(2)_1}2^{\underline{1}}=2\\ ...
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0answers
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The resultant of two homogeneous polynomials is homogeneous

I haven't been able to find a proof for this theorem in the literature: Let $f,g\in k[x_0,\dots,x_k]$ be homogeneous polynomials, of degree $m$ and $n$ respectively. Then $R_{x_0}(f,g)$ is ...
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1answer
93 views

Determining existence of roots of a polynomial in the unit disk (possibly with Rouché's theorem?)

I'm studying for my PhD prelim exam in complex analysis, and I ran into this example problem. Show that the polynomial $$p(z)=z^{47} − z^{23} + 2z^{11} − z^5 + 4z^2 + 1$$ has at least one root ...
1
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2answers
27 views

Polynomials of best approximation

The question is about approximating the continuous function in an interval $[a, b]$. If we consider the linear space of all such functions endowed with the norm $$||f|| = \max_{x \in [a, b]}|f(x)|$$ ...
1
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1answer
12 views

Power with odd base and polynomial having no integral or rational solutions

The initial task is to prove, using induction, that if $b$ is odd and $n\ge1$, then $b^n$ is odd. I think I got that part. Then, using this fact, you have to show that the polynomial $x^{19}+x+1=0$ ...
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45 views

Factorization and Roots of the following Polynomials

I'm struggling with this exercise $(a)$$f:=T^4 +6T^2 -8T - 3 \in \mathbb{Q}[T]$. Show that f is irreducible, with exactly $2$ real roots. $(b)$ Let $\alpha$ und $\beta$ be two real roots of ...
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3answers
778 views

Recursive Integration over Piecewise Polynomials: Closed form?

Is there a closed form to the following recursive integration? $$ f_0(x) = \begin{cases} 1/2 & |x|<1 \\ 0 & |x|\geq1 \end{cases} \\ f_n(x) = ...
10
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2answers
817 views

Algorithm(s) for computing an elementary symmetric polynomial

I've run into an application where I need to compute a bunch of elementary symmetric polynomials. It is trivial to compute a sum or product of quantities, of course, so my concern is with computing ...
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0answers
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Unable to get matched answer using factorization

I have question to solve by factorization. the question is $$(a+b)x^2 + (a+2b+c)x + (b+c) = 0$$ the answer should be $$x = -a, -b.$$ i have done using it \begin{align} (a+b)x^2 + (a+b+b+c)x + ...
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0answers
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Polynomial P= Q² + R² [duplicate]

I want to show that every polynomial $P$ $(\geq0)$ can be written with two other polynomials $Q$ and $R$. $$P=Q^2+R^2$$
11
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2answers
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Solve $x^7-5x^4-x^3+4x+1=0$ for $x$

Solve for $x$ $$x^7-5x^4-x^3+4x+1=0$$ This equation has been bugging me since the past few days. I have found, using the Rational Root Theorem that $x=1$ is a root of this equation. However, ...
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1answer
28 views

product of polynomials in $K[X_1,…,X_n]$ where $K$ is a field.

Let \begin{align*} \Phi_\sigma :E&\longrightarrow E,\\ p(X_1,...,X_n)&\longmapsto p(X_{\sigma (1)},...,X_{\sigma (n)}), \end{align*} where $\sigma \in\mathfrak S_n$. I would like to show that ...
6
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3answers
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Zero divisor in $R[x]$

Let $R$ be commutative ring with no (nonzero) nilpotents. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ is a zero divisor in $R[x]$, how do I show there's an element $b \ne 0$ in $R$ such that ...
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0answers
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A question on composition of polynomial functions

Let $x\in R^n$ and $f(x)$ be any polynomial. Is there a set of polynomial functions: $f_1$, $f_2$,... $f_k$, with each in the form of: $f_i(x) = g_i(x_{1:(n-1)})+h_i(x_n)$ such that $f = f_1\circ ...
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0answers
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Value of K that produces same remainder with two divisors? [closed]

$f(x)=3x^3+6x^2+Kx−4$I am having difficulty finding $K$ such that $f(x)$ has the same remainder when divided by $x-1$ and $x+2. Any help would be greatly appreciated.
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Other method show that $ A(x)=x^2+x+1=0$ has a zeros in $\mathbb{R}$ but why this contradiction?

Let $ A(x)=x^2+x+1$ be a quadratic polynomial equation and $ x \in\mathbb{R}$. It is well known that $ A(x)=x^2+x+1=0$ hasn't a roots in $\mathbb {R}$ , we choose another way to solve this equation ...
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2answers
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Polynomial modulo n

http://en.wikipedia.org/wiki/AKS_primality_test How can I interpret what the "mod n" means? I have watched the Numberphile video on the AKS primality test, and based on that, I am assuming that ...
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1answer
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How to fully factor a polynomial of 4th degree?

How to fully factor this polynomial? $$ 2x^4+3x^3-32x^2-48x$$ Can anyone describe the full steps to factor it? Thanks for the help.
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Long division of $X^3+2X^2+4 $ by $X+2$ produces $0$, why? [closed]

The problem is to divide the polynomials: $$\frac{X^3+2X^2+4 }{ X+2}$$ When I do this, on the second line I get a result of $0$. What did I do wrong?
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1answer
21 views

Finding the value of $y=b^2(3a^2+4ab+2b^2)$ if $a^2(2a^2+4ab+3b^2)=3$ and $a$ and $b$ are distinct zeros of $x^3-2x+c$

If $a$ and $b$ are distinct zeroes of the polynomial $x^3-2x+c$ and $$a^2(2a^2+4ab+3b^2)=3$$ $$b^2(3a^2+4ab+2b^2)=y$$ Evaluate $y$ I tried for many hours but couldn't solve this question. ...
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3answers
45 views

Polynomial interpolation

I need to find the polynomial of degree 3 with respect to these conditions: $ p(0) = 1 $ $ p(1) = -1 $ $ p'(0) = 1 $ $ p''(0) = 0 $ How do I deal with the condition on the second derivative?
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1answer
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Partial fractions decomposition and polynomials

Let $f(z) = a_n z^n + ... + a_0 $ be complex polynomial. we can write $$ f(z) = \prod_{j=1}^k (z-r_j)^{m_j} $$ Why does it follow that $$ \frac{ f'(z)}{f(z)} = \sum_{j=1}^k \frac{m_j}{z-r_j} $$ ...
4
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1answer
59 views

Recurrent problem about polynomials

Given is a sequence of polynomials $P_n$, defined as follows: $P_0(x)=0, P_{n+1}(x) = P_n(x) + \frac{x-P_n^2(x)}{2}. $, n= 0,1,2,..., and x is real. Proving that for all non-negative integers n and ...
2
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2answers
386 views

Factorising polynomials resulting in surds

I am trying to factorise $x^2-18x+60$. Wolfram Alpha tells me this factorises to $(x-\sqrt21-9)(x+\sqrt21-9)$, but what technique should I be using to find this myself?
6
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2answers
80 views

Evaluate this Trigonometric Expression

Evaluate $$ \sqrt[3]{\cos \frac{2\pi}{7}} + \sqrt[3]{\cos \frac{4\pi}{7}} + \sqrt[3]{\cos \frac{6\pi}{7}}$$ I found the following $\large{\cos \frac{2\pi}{7}+\cos \frac{4\pi}{7} + \cos ...
0
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0answers
21 views

$a=3X^2+X+2 \in \mathbb{Z}_7[X]$. Compute the inverse of $[a]$ in $\mathbb{Z}_7[X]/(X^3+4)$

$a=3X^2+X+2 \in \mathbb{Z}_7[X]$. Compute the inverse of $[a]$ in $\mathbb{Z}_7[X]/p(x)$ where $p(x)=X^3+4$ I know that we want some $[b]\in\mathbb{Z}_7[X]/p(x)$ such that $[a][b]=[1]$. I used ...
2
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0answers
27 views

Polynomial Matrix Eigenvalue problem. Conditions under which there are only two complex eigenvalues?

I'm solving the polynomial matrix eigenvalue problem $(A\lambda^2+B\lambda+C)v=0 $. This is what I want the eigenvalues to look like. Are there any conditions on the matrices A,B,C such that there are ...
1
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0answers
63 views

Summation verification

I have a particular polynomial $$ 1-10x+35x^2-50x^3 $$ Which can be written nicely as $$1-(1+2+3+4)x+(1\cdot2+1\cdot3+1\cdot4+2\cdot3+2\cdot4+3\cdot4)x^2$$ ...
0
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0answers
15 views

Are the questions in each of the following sets a family? [closed]

$$Y=(3X+1)(2X-1)(X+3)(X-2)$$ $$Y=2(3X+1)(2X-1)(X+3)(X-2)$$ $$Y=3(3X+1)(2X-1)(X+3)(X-2)+1$$ $$Y=4(3X+1)(2X-1)(X+3)(X-2)+2$$ I am having trouble determining if these functions are all a family. If so, ...