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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for which a, the matrix A is diagonalizable?

A = $ \begin{pmatrix} 2a+3 & 0 & 0 \\ -a-3 & a & a+3 \\ a & a & a+3 \\ \end{pmatrix} $ Characteristic polynomial: $ ...
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0answers
25 views

Derivation of composite Gaussian quadrature error formula

I am working on studying for the Numerical Analysis qualifying exams. One of the questions I am stuck on is the following: Derive the error term for the composite Gaussian quadrature rule with ...
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1answer
42 views

Polynomials - getting wrong answer using Euclidean algorithm

I am finding the GCD of $a = x^3 + 11/3x^2 + 17/4x + 3/2$ and $b = 3x^2 + 22/3x + 17/4$ using the Euclidean algorithm. So I divide $a/b$ and get $q$ and $r$ such that $a = qb + r$. Then, according to ...
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1answer
273 views

$\epsilon>0$ there is a polynomial $p$ such that $|f(x)-e^{-x}p|<\epsilon\forall x\in[0,\infty)$

Could any one tell me how to solve this one? Given $f\in C[0,\infty)$ such that $f(x)\to 0$ as $x\to\infty$ we need to show that for any $\epsilon>0$ there is a polynomial $p$ such that ...
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0answers
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Polynomials with rational zeros

Find all polynomials $F(x)={a_n}{x^n}+\cdots+{a_1}x+a_0$ satisfying $a_n \neq0$; $(a_0, a_1, a_2, \ldots ,a_n)$ is a permutation of $(0, 1, 2 ... n)$; all zeros of $F(x)$ are rational.
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0answers
70 views

Irreducibility of some polynomial

Let $p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1$, for some $k \geq 2$ fixed. I would like to know if $p(x)$ is irreducible in $\mathbb{Q}[x]$.
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1answer
272 views

Using the Multinomial Theorem to Calculate a Finite Sum raised to an exponent

I know it's a simple question, but I keep getting different general formulas for the coefficients when I am trying to use the multinomial theorem for the following: $$ ...
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0answers
39 views

How “separable” (not in that sense) is a polynomial?

Since "separable" is used for different meaning in separable polynomial and separation of variable, I am having trouble searching for anything related to my question. So I hope someone can help with ...
2
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1answer
103 views

Three phase voltage system of polynomial equations

I'm working with the development of a product in the company where I work. This product measures three phase voltages and currents. I cannot change the circuit because it has been sold for a long time ...
0
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3answers
42 views

In $\mathbb{Q}[x]$, what is the gcd of $x^6 − 1$ and $x^4 − 1$…Using Euclid's Method. [closed]

In $\mathbb{Q}[x]$, what is the gcd of $x^6 − 1$ and $x^4 − 1$...Using Euclid's Method. Could someone please do the first few steps of this so I know how to solve gcd of polynomials using Euclid's ...
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1answer
40 views

Efficient Extended GCD Algorithm for Polynomials

For computing the GCD of two multivariate polynomials we have the Euclidian algorithm. However, it's well known that the Euclidian algorithm is not very efficient (because of intermediate expression ...
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0answers
21 views

vector subspace of all real polynomials which are divisible with $x^2 + 1$

Show that the set of all real polynomials which are divisible with $x^2 + 1$ is a vector subspace of space of all real polynomials to 4th degree. Also find base and dimension of this subspace. I ...
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3answers
60 views

complex roots calulation question

How can we find the roots of an equation such as:$z^2 +z +1=0 ,z \in \mathbb{C} $ ?
2
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0answers
111 views

Reduced Gröbner basis and extension of scalars

Consider a field extension $L\subseteq K$, and let $\mathfrak a\neq 0$ be an ideal of the polynomial ring $L[T_1,\ldots,T_n]$. Suppose that a monomial order is fixed, so there exists a unique reduced ...
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2answers
86 views

Find the roots of the equation $(1+xi)^n+(1-xi)^n=0$

Find the roots of the equation $f(x)=(1+xi)^n+(1-xi)^n=0$. I'm having problems finding the roots...this is what I've done: First I expressed $(1+xi)^n$ and $(1-xi)^n$ in trigonometric form and ...
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5answers
69 views

Graphing polynomials

Sketch a graph of the polynomial $P(x)=(x-2)^2(x+1)^3$. You must plot and label the x and y intercepts and these should be the only points you plot. How do I sketch the graph of a polynomial?
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0answers
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Determining if a given equation is solvable given a set of ultra-radicals

So suppose someone is armed with the tools of standard arithmetic, exponents (and of course that comes along with roots) AS WELL AS a set of inverses for some polynomials which are not solvable using ...
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1answer
44 views

About $\mathbb{F}_7[x]$

can you help me with this? Let $a(x)=3x^6+2x^2+x+5$ and $b(x)=6x^4+x^3+2x+4$, find the g.c.d between $a(x)$ and $b(x)$ in $\mathbb{F}_7[x]$. Thanks!
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2answers
50 views

Show that $(x-\alpha)(x-\overline{\alpha})$ is a also a factor of $p(t)$ over the complex numbers

Here is the full question. Lots of struggles: Let $p(t)$ belong to $P(R)$. a) If $(x − \alpha)$ is a factor of $p(t)$ over the complex numbers (i.e. $p(t) = (x − \alpha)\cdot q(t)$, for ...
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1answer
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A bunch of questions involving polynomials.

Okay, so apparently, I can't write more than one post in the space of 20 minutes. So, I'm writing down all the questions I wasn't able to solve here. It would be great if you could solve them and ...
2
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2answers
218 views

Finding the remainder when a polynomial is divided by another polynomial. [duplicate]

Find the remainder when $x^{100}$ is divided by $x^2 - 3x + 2$. I tried solving it by first calculating the zeroes of $x^2 - 3x + 2$, which came out to be 1 and 2. So then, using the Remainder ...
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2answers
46 views

Find the values of a,b and c in a polynomial $p(x) = ax^2 + bx + c$

The question is this : A polynomial $p(x) = ax^2 + bx + c$ where $a,b,c$ are some rational numbers, has $1 + \sqrt3$ as one of the zeroes and also $p(2) = -2$. Find the values of $a,b$ and $c$. ...
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2answers
30 views

Finding a cubic polynomial whose zeroes are the same as collectively of two other quadratic polynomials.

The question is: Find a cubic polynomial $p(x)$ whose zeroes are the same as those collectively of polynomials $g(x) = 2x^2 - 9x + 4$ and $f(x) = 2x^2 + 3x - 2$. Given that $p(0)$ = 8. I tried ...
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1answer
18 views

Finding the remainder polynomial for a given polynomial.

When a polynomial $p(x)$ of degree 3 is divided by $3x^2 − 8x + 5$, quotient and remainder obtained are linear polynomials such that $p(1)$ = 19 and $p(5/3)$ = 25. So, find the remainder polynomial. ...
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2answers
35 views

Real polynomials, complex zeroes and the Intermediate value theorem

I have a second grad polynomial p(x). For arguments sake lets say $$p(x) = x^2 + 16x + 76$$ I also have an inequation $$p(x) > 0$$ Now the inequation does not have a real solution, but only ...
3
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3answers
516 views

What is the lowest-degree function that passes through these points?

I want to find a (preferably polynomial) function that passes through the following twelve points: $(1, 0)$ $(2, 3)$ $(3, 3)$ $(4, 6)$ $(5, 1)$ $(6, 4)$ $(7, 6)$ $(8, 2)$ $(9, 5)$ $(10, 0)$ $(11, ...
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2answers
108 views

Find the value of “k” so that the quadratic polynomial has equal zeroes.

The question is this: Find the the value(s) of $k$ so that the quadratic polynomial $kx^2 + x + k$ has equal zeroes. Answers along with appropriate explanations would be appreciated. Thanks.
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1answer
262 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
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2answers
60 views

Matrix with rank 3 does not exist in this $p(x)$

Given: Characteristic polynomial is $p(x) = x^7 - x^5 + x^3$ . Prove that there isn't a matrix A that $ \rho(A) = 3 $ I tried to play with $p(x) = x^3(x^4 - x^2 +1)$ But I'm still not sure how ...
2
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2answers
52 views

Trying to understand proof that there is always an integer such that a polynomial is composite

I'm trying to follow the main given answer here There is a positive integer $y$ such that for a polynomial with integer coefficients we have $f(y)$ as composite What I don't understand is where in ...
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5answers
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How to prove that a polynomial of degree $n$ has at most $n$ roots?

How can I prove, that a polynomial function $$f(x) = \sum_{0\le k \le n}a_k x^k\qquad n\in\mathbb N,\ a_k\in\mathbb C$$ is zero for at most $n$ different values of $x$? (Except $n=0$ where $f(x)$ is ...
3
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6answers
122 views

Show that all real roots of the polynomial $P (x) = x^5 − 10x + 35$ are negative.

I got this problem out of Andreescu's Putnam and Beyond. I solved it differently from the given solution and could not understand the solution. Can you explain what is happening in the last step of ...
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3answers
54 views

A closed formulae for the coefficient of $x^k$ in $(x-1)^a(x+1)^b$

Let a,b positive integer Do you know any closed formulae for the coefficient of $x^k$ in $(x-1)^a(x+1)^b=\sum_{k=0}^{a+b}u(k;a,b)x^k$ ? I look for an a closed expression of $u(k;a,b)$ involving ...
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6answers
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Simplifying polynomial fraction

Working through an old book I got and am at this problem: Simplify: $$\frac{3x^2 + 3x -6}{2x^2 + 6x + 4}.$$ The answer is supposed to be $\frac{3(x - 1)}{2(x - 1)}$. I thought I had all this ...
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1answer
46 views

Characteristic polynomial $p_{cA}(t)$

Let's define $p_{A}(t)$ the characteristic polynomial of square matrix $A$ over $R$. Prove that for every $c \in R$, $c \ne 0$ the characteristic polynomial $p_{A}(t)$ of the matrix $cA$ is ...
2
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1answer
54 views

Given roots (real and complex), find the polynomial

This is not a duplicate of theory of equations finding roots from given polynomial. Given that the roots (both real and complex) of a polynomial are $\frac{2}{3}$, $-1$, $3+\sqrt2i$, and $3+\sqrt2i$, ...
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2answers
108 views

On factorization of polynomials

I would be very grateful if you give me a hint on problem 9, section 3.6 of Hungerford Algebra, regarding factorization in polynomial rings, saying that: Suppose $f(x)= ...
2
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1answer
57 views

Polynomial equal to the ceiling of x

For a few days, I've been looking for a polynomial who's value is equal to the ceiling function of the only variable it contains. I thought about it for while and I haven't got a clue.
2
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2answers
57 views

Basic irreducible polynomial

I'm studying cyclic codes over a ring $R$. It is well known that a cyclic code over $R$ of length $n$ is an ideal of $R\left[ x \right]/\left( {{x^n} - 1} \right)$. Hence the factorization of ...
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3answers
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Prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$

I want to prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$. Eisenstein's criterion doesn't apply here, so I think the simplest method would be to use the Rational Roots Test, right? If I can use ...
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0answers
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Differential Formula Simplification

Define operators $x,D,1$ by $xf=xf$, $Df=\frac{d}{dx}f=f'$, and $1f=f$. Notice, then that $$(x+D)^nf=\sum_{k=0}^np_k(x)D^kf,\ \ \ \ \ \ \ f\in R[x],$$ for some sequence of polynomials ...
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2nd Order Polynomial Trendline

Hi apologies in advance if this is very trivial or I am out of my depth. I am working on a 2nd Order Polynomial Trendline How do I solve for x here? $$y = (c_2 \times x^2) + (c_1 \times x ^1) + ...
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1answer
23 views

numerical algorithms for determining least common multiple of polynomials

I have a pair of rational polynomial fractions $\frac{A(x)}{B(x)} + \frac{C(x)}{D(x)}$ where A, B, C, and D are all polynomials in x, and I have their coefficients as an array of numbers. I would ...
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Question on Primitive Polynomials

How can we show that any $g(x)\in \mathbb{Q}[x]$ can be uniquely written as $g(x)=cf(x)$ where c is rational and $f\in\mathbb{Z}[x]$ is primtive? This property seems intuitive but I am unable to ...
0
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1answer
30 views

What do I do wrong with Möbius method of inversion?

I use the Möbius inversion with polynomials as e.g. in the well-known inversion formula of the cyclotomic polynomials. So I have $$p_{2n}(x)=\prod_{d|n}(2q_d(x))^{\mu(\frac{n}{d})}$$ Now I get the ...
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4answers
83 views

Sums of solutions to $z^n-1 = 0$ that equal 0

Consider the solutions of the equation $z^n - 1 = 0$, where $z$ is a complex number: ${z_1,z_2...z_n}$. What are ALL the possible sums $\sum_{i=1}^n a_iz_i$ over these n solutions, where $a_i$ are ...
0
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1answer
34 views

Complex conjugate root theorem question

From the Complex conjugate root theorem we get that if a polynomial in one varaible with real coefficients has as solution $a + bi$ , than $a-bi$ must also be a solution...however, what happens if ...
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2answers
26 views

Polynomial representation of binary

It is well known that we can represent binary using polynomial. For example, $11$ can be represented as $x+1$. So when we compute $11\times11$, we should obtain $1001$, which is equal to $9$ in ...
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4answers
71 views

Prove that the characteristic polynomial of a nilpotent matrix is $x^n$

How can I prove that the char.pol. of a nilpotent matrix is of the form $x^k$? I'm trying to do it by contradiction but assuming that $p_{xA}=a_0+a_1x+\dots+a_mx^m+\dots+a_nx^n$ seems not giving any ...
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2answers
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Dimension of quotient construction

If I have an irreducible polynomial, $f$ with $deg(f) = n$ and I look at the quotient: $$R = \frac{\mathbb{Q}[x]}{(f)}$$ How can we show that the dimension of $R$ as a $\mathbb{Q}$ vector space is ...