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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4
votes
3answers
144 views

Closed form of a sum of binomial coefficients?

I have the following function: $T_n(d)=\sum\limits_{k=\frac{n-d}{2}}^{\lceil \frac{n}{2} \rceil}{k\choose \frac{n-d}{2}}$ ${n \choose 2k}$, where $n,d\in \mathbb{N}^0$, and $n,d$ have the same ...
0
votes
0answers
15 views

factorisation over a galois field

I got a question about two examples in my studybook about the factorisation of a galois field. I have included a screenshot of both my examples along with some clarification as it's written in Dutch, ...
3
votes
5answers
118 views

difference between the polynomials

I have a homework assignment that I do not know how to solve. I don't understand how to calculate $f(x)$ in this assignment. $f(t)$ is the difference between the polynomials $2t^3-7t^2-4$ and ...
1
vote
2answers
53 views

Find a generator for an ideal in $\mathbb{Q}[T]$

Let $I$ be the ideal in $\mathbb{Q}[T]$ generated by $L=\{T^{2}-1, T^3-T^2+T-1,T^4-T^3+T-1\}$. Find $f\in\mathbb{Q}[T]$ such as $(f)=f\mathbb{Q}[T]=I$. The book solution proves that $I\subseteq ...
0
votes
0answers
34 views

existence of a positive root

Consider the polynomial $$ P(\omega)=\omega^8+\phi_7\omega^7+\phi_6\omega^6+\phi_5\omega^5+\phi_4\omega^4+\phi_3\omega^3+\phi_2\omega^2+\phi_1\omega+\phi_0 $$ with real coefficients. Assuming that ...
6
votes
2answers
218 views

Complex numbers system of equations problem with 5 variables

Let $z_0$,$z_1$,$z_2$,$z_3$ and $z_4$ such that $z_i\in C$ that hold: $$(1)|z_0|=|z_1|=|z_2|=|z_3|=|z_4|=1$$ $$(2)z_0+z_1+z_2+z_3+z_4=0$$ $$(3) z_0z_1+ z_1z_2+z_2z_3+z_3z_4+z_4z_0=0$$ Prove that ...
0
votes
0answers
11 views

Polynomial modulus in Quotient Ring

I have a ring $R=\Bbb Z[x]/(x^m+1)$ with $m$ some power of two and a polynomial $g \in R$, which has relatively small coefficients and some other properties that I believe to be irrelevant for this ...
2
votes
0answers
6 views

Algorithms for solving overdetermined, homogeneous linear systems with multivariate polynomial coefficients

I would like to solve overdetermined, homogeneous linear systems of equations with multivariate polynomial coefficients, i.e., $Ap=0$ with $A$ an $m\times n$ matrix, $m\gg n$, and $a_{i,j} \in ...
5
votes
0answers
167 views

How control small perturbations keeping a zero of a polynomial?

Let $P : \mathbb{R}^n \to \mathbb{R}^m$ by a polynomial function, with $n \ge m$. Let $x_0 \in \mathbb{R}^n$ be a zero of $P$ such that the graph of $P$ crosses $\mathbb{R}^n \times \{ \vec{0}\}$ ...
11
votes
4answers
239 views

Which polynomials fix the unit circle?

Find all polynomials $P(x)$ with real coefficients such that for every $x,y\in \mathbb{R}$ satisfying $x^2+y^2=1$ we have $$P(x)^2+P(y)^2=1$$
-7
votes
0answers
43 views

Solving a system of polynomial equations in three variables (x^2-yz=18, y^2-zx=8, z^2-xy=-7)

Solving a system of polynomial equations in three variables (x^2-yz=18, y^2-zx=8, z^2-xy=-7 I've tried rearranging each equation to isolate for one variable ex: z^2-xy=-7 --> z= x^2-18/y after, I ...
8
votes
3answers
410 views

Every time a real solution.

I have got an interesting exercise. Proof that for all positive integer $a$ and $p(x) = x^2+2013x + 1$, $\underbrace{p(p(\dots p}_{a \ \ \text{times}}(x)\dots )) = 0$ has got at least 1 real solution ...
1
vote
3answers
36 views

Find isomorphism between $\mathbb{Q}[T]/(T^2+3)$ and $\mathbb{Q}[T]/(T^2+T+1)$

The books states that the isomorphsim is $g(T)=2T+1$ and the identity when restricted to $\mathbb{Q}$. I would like some help to understand what the process is to find $g$.
3
votes
1answer
19 views

System of quadratic equations that is symmetric

Solve for $z$: $z^2-3z+1=x, x^2-3x+1=z$ I see that it is symmetric, but not anything else. Hints would be great, but please do not spoil the answer. Thanks!
0
votes
1answer
25 views

Use $\sin^22t=4\sin^2t(1-\sin^2 t)$ to show that $\sin t$ is not a polynomial?

I am reading Barbeau's Polynomials and I found the following problem: Use the identity $\sin^22t=4\sin^2t(1-\sin^2 t)$ to show that $\sin t$ is not a polynomial. But I really have no idea on how ...
4
votes
2answers
1k views

What is the algorithm for long division of polynomials with multiple variables?

I was helping a high-school student last night whose teacher had given as a homework problem the division $$\frac{15x^4-y^2}{x^2+y};$$ I tried a heuristic involving splitting off a difference of ...
2
votes
1answer
41 views

Silly number theory questions I can't prove.

I know if $gcd(r,s)=1$ then $1=as+bs$ for some intgers $a,b$. Here's what I want to know: which numbers can be written as $as+bs$, if I am restricted to $a,b \in \mathbb{N}$? To be more specific, I ...
-3
votes
1answer
40 views

Find $g(x)$ if $(x^2+a^2)(x^2 + b^2)(x^2 + c^2) = (f(x))^2 + (g(x))^2$ and $f(x)$ is a degree three polynomial [on hold]

If $$(x^2+a^2)(x^2 + b^2)(x^2 + c^2) = (f(x))^2 + (g(x))^2$$ where $f(x)$ is a degree three polynomial, find $g(x)$.
0
votes
2answers
53 views

Can someone help me to prove this theorem from Axler's *Linear Algebra Done Right*?

If $p\in P(\Bbb{R})$ is a nonconstant polynomial, then $p$ has a unique factorization (except for the order of the factors) of the form ...
0
votes
1answer
17 views

Different forms of a quadrature

I am solving the following problem: Find the quadrature of the following form: $Q(f) = Af(−1) + Bf(0) + > Cf(1)$, which has the highest degree and interpolates the integral: $\int_{-3}^{3} ...
5
votes
2answers
5k views

Reed Solomon Polynomial Generator

I am developing a sample program to generate a 2D Barcode. And i am using reed solomon error correction code. By Going through this article i am developing the program. But i couldn't understand how ...
0
votes
2answers
162 views

Graeffe's root finding method

What are the practical applications of Graeffe's root finding method?I searched a lot but couldn't find.I found that it is used in aerodynamics and electric circuit analysis.But don't know much about ...
1
vote
0answers
35 views

Irreducible polynomial

Does there exist an irreducible polynomial over a field K with two roots $a,b$ and $k\in K$ such that $a=b+k$ ? This can't happen if K is of characteristic $0$ , but can it happen if K is of ...
1
vote
0answers
45 views

Discriminant of $4g+h^2$

Suppose we have two polynomials $g,h\in \mathbb Z[x]$ with $\deg g = 2k+1 =:n$ and $\deg h=k$. As an example, take $g=x^7+2x^6+x+2$, $h=x^3+x+7$. My question is: Why does the discriminant of ...
2
votes
0answers
36 views

the numer of monic irreducible polynomials of degree $3$ in $\mathbb{F}_q$

I want to know how hany monic irreducible polynomials of degree $3$ there are in a field $\mathbb{F}_q$. The whole number of monic polynomials of degree three is $q^3$. Now I want to find out how ...
1
vote
1answer
43 views

Chinese remainder theorem for polynomial evaluation

Let $R$ be a euclidean domain, $m_0,\ldots ,m_{k-1}\in R$ be pairwise coprime and $m:=m_0\cdots m_{k-1}$. The Chinese remainder theorem states: $$\varphi:R\to R/(m_0)\times\cdots \times ...
0
votes
1answer
21 views

Proof the Existence and Uniqueness of Factorization Form of Polynomial with Complex Coefficient

If $p\in P(\Bbb{C})$ is a nonconstant polynomial, then $p$ has a unique factorization (except for the order of the factors) of the form $$p(z)=c(z-\lambda_1)....(z-\lambda_m)$$ where ...
2
votes
3answers
46 views

How to multiply the binomials $(2x^3 - x)\left(\sqrt{x} + \frac{2}{x}\right)$

I am sorry if the numbers are not formatted, I have searched but found nothing on how. I am trying to multiply $$(2x^3 - x)\left(\sqrt{x} + \frac {2}{x}\right)$$ together and I arrive at a different ...
4
votes
5answers
440 views

Polynomials Shouldn't Have factors using Rational Root Theorem but it does!

I came across this polynomial $X^4 + X^3 + 2X^2 + X + 1$ I tried to factor it using Rational root theorem, but it seems there are no roots possible. 1 or -1 don't work. But I know for a fact ...
1
vote
0answers
18 views

Derivatives of Lagrange polynomials

It seems there is some relationship between Lagrange polynomial and Legendre polynomial. That is Lagrange polynomial can be expressed as a function of Legendre polynomial. If so, I could use this ...
2
votes
1answer
105 views

Polynomial of matrix

The question here is that, is it possible to solve a polynomial of matrix like the following $A^{2}+A=B$, where $B$ is a known semi-definite matrix, and $A$ is the unknown symmetric matrix we ...
3
votes
0answers
25 views

If a Sequence of Polynomials Converge to Another Polynomial Then the Roots Also Converge.

Proposition 5.2.1 in Artin states that: THEOREM. Let $p_k(t)\in \mathbf C[t]$ be a sequence of monic polynomials of degree $\leq n$, and let $p(t)\in \mathbf C[t]$ be another monic polynomial ...
4
votes
3answers
254 views

How can I show why this equation has no complex roots?

I've been asked to show why an equation has no complex roots but i'm at a complete loss. The equation is $F_{n+2}=F_n$ Where $F_n=(x-1)(x-2)...(x-n)$ and n is a positive integer. I'd really ...
0
votes
1answer
49 views

Find solutions to given equation

Find all integer solutions $x$ for $0 < x < 10^9$ of the equation: $$x=b\cdot s(x)^a+c,$$ where $a$, $b$, $c$ are some predetermined constant values and function $s(x)$ determines the sum of ...
3
votes
2answers
50 views

Show that if $\mathrm{Tr}(y)=0$ then there exists a $x$ such that $x^p-x=y$.

We have the Trace map defined by: $$ \mathrm{Tr}\colon \mathbb{F}_q\rightarrow\mathbb{F}_q\colon x\mapsto x+x^p+x^{p^2}+\cdots+x^{p^{n-1}}, $$ where $q=p^n$. Now I have to prove that if ...
15
votes
3answers
2k views

Shortcut/trick for integrating a factored polynomial?

If I'm integrating a factored polynomial, say $$\int{x(x+1)(x-2)(x+3)dx},$$ does some shortcut exist that keeps me from having to expand the polynomial? Currently, I'd just do all the multiplication ...
29
votes
7answers
2k views

Using Gröbner bases for solving polynomial equations

In my attempts to understand just how computer algebra systems "do things", I tried to dig around a bit on Gröbner bases, which are described almost everywhere as "a generalization of the Euclidean ...
0
votes
1answer
40 views

Understanding edge correction with a 2nd order polynomial in Gaussian filter

I am trying to understand the following code from ImageJ: http://pastebin.com/tXfhNxqf The problem: When computing the gaussian kernel we use the gaussian function $$ f(x) = e^{-\dfrac{x^2}{2 ...
1
vote
1answer
541 views

Relative Maxima/Minima of polynomial functions

I am taking the Pre Calculus 12 course online. I came across this concept that the online material teaches in 3 different ways, and each one contradicts the other. I find this extremely frustrating. ...
0
votes
0answers
48 views

Polynomial of Degree 3 Solutions [duplicate]

If $p(x) \in F[x]$ is of degree $3$, and $p(x)=a_0+a_1x+a_2x^2+a_3x^3$, show that $p(x)$ is irreducible over $F$ if there is no element $r\in F$ such that $a_0+a_1r+a_2r^2+a_3r^3 =0$. If $p(x)$ is ...
-3
votes
1answer
29 views

Polynomials roots and coefficients [on hold]

The equation $x^ 3 +x−1 = 0$ has roots $\alpha, \beta, \gamma$. Show that the equation with roots $\alpha^3, \beta^3, \gamma^3$ is $y^3−3y^2+4y−1 = 0$. Hence find the value of $\alpha^6 +\beta^6 ...
0
votes
0answers
13 views

Find the polynomial interpolating function $f(x)=\cos\left(\frac{\pi}{2}x\right)$ [closed]

Find the polynomial interpolating function $f(x)=\cos\left(\frac{\pi}{2}x\right)$ at points: $\{-1,0,1,2\}$ Write this polynomial as Lagrange, Newton and power polynomial.
1
vote
1answer
24 views

Finding polynomial optimal in terms of least squares approximation

Find polynomial $w$ of degree at most $2$ optimal in terms least squares approximation for a function $f(x)=x^3$ in the norm $\|g\|=\sqrt{(g,g)}$, given that: $$ (f,g) = \int\limits^1_0 f(x)g(x)dx. ...
2
votes
1answer
45 views

Calculating Vandermonde determinant

I understand that the Vandermonde determinant $$ W(x_1, \ldots, x_n) = \left| \begin{array}{cccc} 1 & 1 & \cdots & 1\\ x_1 & x_2 & \cdots & x_n \\ x_1^2 & x_2^2 & ...
0
votes
0answers
11 views

Polynomial Data Fitting with Two Variables

I have the following data and I want to find the equation interpolating the given data. For example equation for given sample data is simple $a^2*b+10$ ...
6
votes
5answers
316 views

How to find the polynomial such that …

Let $P(x)$ be the polynomial of degree 4 and $\sin\dfrac{\pi}{24}$, $\sin\dfrac{7\pi}{24}$, $\sin\dfrac{13\pi}{24}$, $\sin\dfrac{19\pi}{24}$ are roots of $P(x)$ . How to find $P(x)$? Thank you very ...
0
votes
3answers
767 views

How to show that $\{t, \sin t , \cos 2 t , \sin t \cos t \}$ is a linearly independent set of functions on $\mathbb{R}$?

I have this homework question that I have no idea how to do: Show that $\{t, \sin(t), \cos(2t), \sin(t)\cos(t) \}$ is a linearly independent set of functions defined on $\mathbb{R}$. Start by ...
2
votes
1answer
77 views

How prove this polynomial $p(x)$ is deg greater than $n-1$

Question: Let $P(x)$ be a polynomial satisfying $$P(k)=\cos{\dfrac{2k\pi}{n}},k=1,2,\cdots,n$$ Show that $$\deg{P(x)}\ge n-1$$ I want to consider ...
0
votes
1answer
34 views

Finding complex roots of integer polynomials

How would one find approximates for complex root of polynomial with integer coefficients,I know for example the Newton's method $$x_n=x_{n-1}-\frac{f(x_{n-1})}{f'(x_{n-1})}$$ Anyway is it possible to ...
1
vote
1answer
252 views

Wolfram Mathematica - Newton Backward Interpolation?

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 + ...