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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20 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
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5answers
129 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 ...
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0answers
36 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 ...
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2answers
58 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
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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 ...
3
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0answers
14 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 ...
1
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3answers
37 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 $g$.
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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 ...
3
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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!
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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 ...
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3answers
412 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 ...
2
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1answer
42 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 ...
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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 $g(x)$.
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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 ...
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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 ...
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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} ...
2
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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 ...
0
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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 ...
11
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4answers
241 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 $$P(x)^2+P(y)^2=1$$
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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 ...
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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
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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 ...
3
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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
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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 ...
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 ...
-3
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1answer
29 views

Polynomials roots and coefficients [closed]

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 ...
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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 ...
0
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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
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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. ...
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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 ...
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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 $a^2*b+10$ ...
2
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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
1answer
35 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 ...
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 ...
2
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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 ...
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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 ...
0
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0answers
32 views

Is this polynomial equation solvable? $ \alpha x^{n+2} + \beta x^{n+1} + \gamma x^3 + \delta x^2 + \epsilon x + \zeta = 0 $

I have an equation I wish to solve. I was going to solve it numerically but maybe there is a way to handle it analytically? $ \alpha x^{n+2} + \beta x^{n+1} + \gamma x^3 + \delta x^2 + \epsilon x + ...
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0answers
15 views

Characterization of ideals generated by homogeneous polynomials in terms of $f^{(d)}$ in Gathmann's notes.

On pg. 37 of Gathmann's Algebraic Geometry notes, the following is mentioned: For every $f\in k[x_0,x_1,\dots,x_n]$ be an ideal. The following are equivalent: I can be generated by ...
2
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2answers
51 views

Finding a condition on the real $a$ such that $P$ is divisible by $(x-a)^2$

Let $P(x)=\frac{x^3}{6}+\frac{x^2}{2}+x+1$. I have to find a condition on the real $a$ such that $P$ is divisible by $(x-a)^2$. I tried to use Polynomial long division and solve a system (we need ...
0
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1answer
36 views

Notation for vector space of polynomials of bounded degree

Is there standard notation for the vector space of polynomials in $n$ variables with coefficients in a field $F$ and with degree at most $D$? Without bounding the degree, it is $F[x_1, \ldots, x_n]$. ...
0
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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 ...
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0answers
22 views

Question about valuation rings of a rational function field

Let $k$ be a field and $f \in k[x,y] \setminus k$. Write $E=k(f)$ and $L=k(x,y)$. Assume that there exists a $g \in k(x,y)$ such that $L = E(g)$. Suppose there exists a valution ring $\mathcal{O}$ ...
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1answer
17 views

Is a function that has maximum points a polynomial function?

Just like the title says. Is a function that has maximum/minimum points guaranteed to be a polynomial function? Is there any occasion that it cannot be expressed as a polynomial function?
0
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1answer
26 views

Fast Fourier Transformation: inverse transform of the product of polynomials

I have managed to implement and understand most of the Fast Fourier Transformation. However, I have one last question. If one has two polynomials, say $A(x)$ and $B(x)$, and one computes DFT of ...
0
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0answers
7 views

Convex combination of polynomials with roots on the unit circle and companion matrix

Given two $N^{th}$ order polynomials $P_0(z)$ and $P_1(z)$, let their roots be $w_k$ and $z_k$ respectively. All the roots of both polynomials lie on the unit circle $\mathcal{U}$. Also $w_i \neq z_j$ ...
0
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4answers
66 views

What happens if the coefficients of polynomials are not taken from a field of real numbers?

I saw in my abstract algebra textbook that defines the gcd of a polynomial over a field (i.e. the coefficients of the polynomial is taken from a field). My question is that what happens if the field ...
5
votes
3answers
73 views

Existence of polynomial such that $P_n(cos\theta)=cos(n\theta)$

Is there a way of proving existence of a polynomial $P_n(x)$ such that $\cos{(n\theta)}=P_n(\cos{\theta})$ without knowing the Chebyshev polynomials a priori?
0
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0answers
19 views

Fast Fourier Transformation [duplicate]

[Note: I asked this question earlier, but I got no answers. Thus, I am "bumping" it.] I was looking at the Fast-Fourier Transformation today, on this site [if you cannot read Russian, simply use ...
1
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0answers
34 views

Fast Fourier Transformation, explanation of $A(x)=A_0(x^2)+x A_1(x^2)$

I was looking at the Fast-Fourier Transformation today, on this site [if you cannot read Russian, simply use Google Translate, which is what I am doing right now]. http://e-maxx.ru/algo/fft_multiply ...
2
votes
1answer
80 views

How to find all complex polynomial $f$ such that $1+f(z^n+1)=(f(z))^n$

Question: Let $n\gt1$ be a natural number. Is there a non-constant complex polynomial $P$ such that $P(x^n+1)=P(x)^n-1$ for all $x$? I saw this problem about polynomial, here is the question: Find ...