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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Irreducible polynomials help

$f(x)=x^4-16x^2+4$, the root of $f(x)$ is $a= \sqrt{3} + \sqrt{5}$ Factorise $f(x)$ as a product of irreducible polynomials over $\mathbb{Q}$, over $\mathbb{R}$ and over $\mathbb{C}$. I am really ...
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Is the set of polynomials of degree less than or equal to $n$ closed?

This question is in relation to the space $C(I)$, $I = [a, b]$. Define $P_n =\{ a_0+\dots+a_nx^n \mid a_i \in \mathbb{R}\}$ (any or all $a_i$ could be zero); clearly $P_n \subset C(I)$. The norm I'm ...
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Proof by “infinite induction”

Prove that $\sum_{i=1}^{n} i^3 = \left( \frac{n(n+1)}{2} \right)^2$. We can check this is true for n=0,1,2,3,4. Since the right side is a polynomial of degree 4, and the left side is a sum of ...
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How to find the root of a polynomial

I don't know how to solve the following equation: $x^5-h_1x^4-h_2x^3-h_5=0$, where $h_1=\beta_1+\beta_2$, $ h_2=\beta_1+\beta_2-\beta_1\beta_2-\frac{\beta_1\beta_2}{\beta_1+\beta_2}$, ...
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1answer
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The roots of $x^3+4x-1=0$ are $a$, $b$, $c$. Find $(a+1)^{-3}+(b+1)^{-3}+(c+1)^{-3}$

This is a question in A level Further Pure mathematics pastpaper Nov 2010. The roots of the equation $x^3+4x-1=0$ are $a$, $b$ and $c$. i) Use the substitution $y=1/(1+x)$ to show that the equation ...
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1answer
153 views

Is there a reason why the $S$-polynomial is defined in this way?

In my book the $S$-polynomial of two nonzero polynomials $f$ and $g$ is defined as $$S(f,g) = \displaystyle\frac{x^w}{LT(f)} \cdot f - \frac{x^w}{LT(g)} \cdot g$$ where $\displaystyle x^w$ is the ...
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Show set of polynomials make up a linear space basis

How do we show that the set $B=\{1,x+1,x(x+1),x(x+1)(x-1)\}$ is a basis for a linear polynomial space? I know that we need to do two things: Show that our set is a spanning set, meaning that every ...
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1answer
85 views

Describe invariant polynomials under action of commutative group of order eight.

I believe the question below should be fairly standard in invariant theory ; I hope someone more familiar with it than me can explain a bit more or point to a reference. Let $F$ be polynomial field ...
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2answers
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Find the inverse of $\alpha^{38}$ in $\mathbb F = \mathbb Z_2[x]/\left<x^4+x+1\right>$

Let $\alpha$ be a root of $x^4+x+1$ and we are given some powers of $\alpha$ as linear combinations of $1,\alpha,\alpha^2$ and $\alpha^3$ $\alpha^4=\alpha+1$ $\alpha^5=\alpha^2+\alpha$ ... (the rest ...
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Relation between torsion subgroup of multiplicative group of field and solvability of polynomials

In a broad sense, what relationships are there between the torsion subgroup $G$ of the multiplicative group of non-zero elements of a field $K$ and whether or not certain polynomials in $K[x]$ have ...
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2answers
112 views

Encode the message $[1,1,0,1,1,0,1]$ in BCH code based on the field $\mathbb F = \frac{\mathbb Z_{2}[x]}{x^4+x+1}$

So here's what I understand so far: $\mathbb F = \frac{\mathbb Z_{2}[x]}{x^4+x+1} = GF(16)$ The code is written as $[x^{14},x^{13},x^{12},x^{11},x^{10},x^{9},x^{8}$ $|$ ...
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1answer
2k views

How to write a polynomial in standard basis?

How does one write the polynomial $p(x)=\frac{1}{2}x^3+(-\frac{3}{2})x^2+1$ using the standard basis $\{1,x,x^2,x^3\}$ ?
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273 views

Sequence of functions that converge on absolute value

Are there sequences of (real) polynomials ($p_n$) that converge on the absolute value: $\lim\limits_{n\rightarrow\infty} p_n(x) = |x|$? If so, what is it/are they? if not, why not?
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Polynomial proof

This is another problem from I.M Gelfand's book that I am stuck with. Problem 169. Assume that $x_1,\ldots,x_{10}$ are different numbers, and $y_1,\ldots,y_{10}$ are arbitrary numbers. Prove that ...
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122 views

Factorise $f(x)=x^5-x$ into a product of irreducibles in $\mathbb Z_{5}[x]$

So plugging in $1$ gives $f(1) = 0$ which means $1$ is a root and $f$ has a factor $(x-1)$ which is $\equiv (x+4)$ in $\mathbb Z_{5}[x]$ ? I then divide $f(x)$ by $(x+4)$ using polynomial long ...
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6answers
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How to find the roots of $x^4 +1$

I'm trying to find the roots of $x^4+1$. I've already found in this site solutions for polynomials like this $x^n+a$, where $a$ is a negative term. I don't remember how to solve an equation when $a$ ...
3
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1answer
297 views

basis for the space of polynomials $p(x)$ of degree at most $n$ such that $p(2)=p(4)=2p(7)$.

The problem: find the dimension and some basis for the space of polynomials $p(x)$ of degree at most $n$ such that $p(2)=p(4)=2p(7)$. It is easy to see that any polynomial $p(x)$ of degree $n$ such ...
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0answers
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Let $f(x+1) = P(f(x),x)$ where P is a polynomial. Express $f(x)$ as an integral.

Let $x$ be a real number and $f(x)$ a real analytic function such that $f(x+1) = P(f(x),x)$ where $P$ is a given real polynomial. Express $f(x)$ as an integral from $0$ to $\infty$. As an example we ...
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1answer
239 views

f(g(x)) has a degree divisible by n

Let $f(x)$ a irreducible polynomial of degree $n$ over a field $F$. Let $g(x)$ be a polynomial in $F[x]$. Prove that every irreducible factor of the composition $f(g(x))$ has a degree which is ...
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2answers
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How to find the inverse of $y=x^3-5x^2+3x+c$

I know how to find the inverse of $y = x^3$, but once you add/subtract another term, that is where I become lost. I have used wolframalpha to get an answer. I am lost upon how this was obtained. ...
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71 views

what's the inverse of $1 + \alpha$, where $\alpha$ is a root of $p(x) = x^3 + 9x + 6$

Show that $p(x) = x^3 + 9x + 6$ is irreducible in $\mathbb Q[x]$. Let $\alpha$ be a root of $p(x)$. Find the inverse of $1 + \alpha$ in $\mathbb Q[x]$. So as far as the irreducibility is conccerned ...
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3answers
225 views

Find a polynomial only from its roots

Given $\alpha,\,\beta,\,\gamma$ three roots of $g(x)\in\mathbb Q[x]$, a monic polynomial of degree $3$. We know that $\alpha+\beta+\gamma=0$, $\alpha^2+\beta^2+\gamma^2=2009$ and ...
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2answers
89 views

Simple polynomial proof

this problem is from I.M Gelfand's book called Algebra. Problem 165. Assume that $$\left\{\begin{array}116a+4b+c=0\\49a+7b+c=0\\100a+10b+c=0\end{array}\right.$$ Prove that $a=b=c=0$ I know I ...
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3answers
40 views

What are all the elements of $\mathbb Z_{2}[x] / <x^3+x+1>$

List all the elements of $\frac{\mathbb Z_{2}[x]}{<x^3+x+1>}$ (the set of remainders) Please verify my understanding: Since the polynomial is of degree 3, the remainders must have degree at ...
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1answer
52 views

How to get the equivalence classes for $m(x) = x^2+1$ in $\mathbb Z_{2}[x]$?

For $m(x) = x^2+1$ in $\mathbb Z_{2}[x]$, we have $$\frac{\mathbb Z_{2}[x]}{ \langle x^2+1 \rangle} = \{0, 1, x, x+1\}$$ How do we get that set? I think it's supposed to be a set containing all ...
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1answer
91 views

Bernoulli Polynomials

I am having a problem with this question. Can someone help me please. We are defining a sequence of polynomials such that: $P_0(x)=1; P_n'(x)=nP_{n-1}(x) \mbox{ and} \int_{0}^1P_n(x)dx=0$ I need to ...
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1answer
76 views

Are the following linear maps continuous ?

I am supposed to find the whether the following maps are continuous or not , if continuous then to find the $||T||$ $P$ is a vector space of polynomials . Define norm on the polynomials $p\in P$ as ...
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1answer
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factorization of the cyclotomic $\Phi_n(x)$ over $\Bbb F_p$

One can prove that the $\Phi_n(x)$ are irreducible over $\Bbb Z$. Where $\Phi_n(x)=\prod _{(a,n)=1}\zeta_n^a$ (i.e the product of the primitive n-rooth of unity). I want to find a factorization of ...
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1answer
45 views

In $\mathbb R$ is everything a unit and associated with each other?

Let $R$ be an integral domain with identity. A unit of $R$ is an element $u \in R$ which divides 1. Does this mean every element in $\mathbb R$ (real numbers) is a unit since every element divides ...
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91 views

Factor $x^3-x^2+2$ in $\mathbb Z_{3}[x]$

Factor $x^3-x^2+2$ in $\mathbb Z_{3}[x]$ and explain why the factors are irreducible. So the factor is supposed to be: $x^3-x^2+2 = 2(x + 1)(2x^2 + 2x + 1)= (x + 1)(x^2 + x + 2)$. But I don't ...
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Another GCD question: $(x^2-x+1)$ and $(x^3-x^2+2)$

So according to WolframAlpha the answer should be $1$ (And by inspection it certainly looks like it should be $1$. My current working using Euclidean algorithm and polynomial long division ...
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1answer
101 views

Find monic gcd($x^4+x^3+x+1$, $x^6+x^5+x^4+x+1$) in $\mathbb Z_{2}$

My working so far using the Euclidean algorithm and polynomial long division (which I won't fully show here) $x^6+x^5+x^4x+1$ = $(x^2+1) \times (x^4+x^3+x+1) + (-2x^3-x^2)$ and $(-2x^3-x^2) \equiv ...
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112 views

A question regarding the quotient map in a polynomial ring over a field

Let $F$ be a field, $F[X]$ the polynomial ring in one variable and $I$ an ideal of $F[X]$. Then does the quotient map $\pi:F[T]\longrightarrow F[T]/I$ map prime ideals to prime ideals?
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1answer
267 views

Automorphisms of a splitting field

Can someone please tell me how to determine all $\mathbb{Q}$-automorphisms of some splitting field $F$ of $X^3-3 \in \mathbb{Q}[X]$ and to determine $[F:\mathbb{Q}]$ ? I think $[F:\mathbb{Q}]=3$, ...
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1answer
101 views

the minimal polynomial is the characteristic polynomial of a suitable map

How would I have to go about proving that the minimal polynomial $p_\beta$ of a field extension $F\subseteq G$ coincides (modulo the sign) with the characteristic polynomial of the linear mapping ...
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159 views

Cyclotomic Polynomial of a Prime

I have this question on a homework sheet: Claim:$$\Phi_{p}(x)=1+x+x^2+...+x^{p-1}\space$$ for $p$ prime. which was followed by the claim that $\Phi_{p^n}(x)=\Phi_p(x^{p^{n-1}})$ which I have ...
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Polynomial uniqueness proof

this problem is from my Gelfand's Algebra book. Problem 164. Prove that a polynomial of degree not exceeding 2 is defined uniquely by three of its values. This means that if $P(x)$ and $Q(x)$ are ...
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The degree of a polynomial which also has negative exponents.

In theory, we define the degree of a polynomial as the highest exponent it holds. However when there are negative and positive exponents are present in the function, I want to know the basis that we ...
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3answers
742 views

How do I prove $F(a)=F(a^2)?$

Let $E$ be an extension field of $F$. If $a \in E$ has a minimal polynomial of odd degree over $F$, show that $F(a)=F(a^2)$. let $n$ be the degree of the minimal polynomial $p(x)$ of $a$ over $F$ and ...
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Power functions and parabola issue

With the function f(x)=x^2 we get a graph like so... The rule for power functions, that I've been told, is the larger the power gets, the closer the line will touch the x-axis. Example for ...
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1answer
55 views

Need help on dividing polynomials

I have trouble understanding how to divide $x^4 + y^4$ by $f_1 = x^2 + y$ and $f_2 = x^2 y + 1$ using the ordering $ y \leq x$ and separately for $ x \leq y$. Please help! I went to the tutoring ...
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Cyclotomic Polynomials and Euler's Totient Function

Claim: $$\prod_{d \mid n} \Phi_{d}(x) = x^n - 1 $$ Where $\Phi_{m}(x)$ is the $m^{th}$ cyclotomic polynomial of $x$. I think it has to do with Euler's Totient Function $\phi$ and the result $$\sum_{d ...
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1answer
260 views

Factoring polynomials with prime discriminant

I was busy doing a homework exercise in which I had to compute the discriminant $\Delta(f)$ of the polynomial $$f(X) = X^4+X^2+X+1$$ which turned out to be the prime $257$. Subsequently, I was asked ...
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2answers
152 views

Maple help needed

Consider the multivariable polynomial $$g(x,y,z,w)=a_1xyw+a_2xy^2+a_3xyz+a_4x^3z+a_5z^3+a_6y^2z+a_7w^4\;,$$ where $a_1,\cdots, a_7$ are constants. I would like to use Maple to extract the coefficients ...
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Vague question about polynomials and symmetry

The problem to find polynomials $f(x,y)$ such that $f(x,y) - f(y,x) = 0$ can be 'solved' by characterizing the solutions as all polynomials in $xy$ and $x+y$ (according to the fundamental theorem of ...
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37 views

Does an irreducible polynomial in K(t)[x] give an irreducible polynomial in K[t][x]

Let $K[t]$ be the ring of polynomials over a field $K$. Let $K(t)$ be its fraction field. Let $f$ be an irreducible polynomial in $K(t)[x]$. There exists an element $a\in K[t] $ such that $af$ is in ...
2
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2answers
2k views

Construct a polynomial function with the given graph

Where does one begin? I can see that the zeros are -5, -3, 0, and 4? Is that correct so far?
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162 views

How to validate this Matlab code?

I am new to Matlab, help please. In the book I saw this picture for the following polynomial: $$g(t)=x_1+x_2t+x_3t^2+ \ldots +x_{10}t^9$$ for some $t$ and $g(t)$. The following code is given: ...
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1answer
40 views

Is $ U = \{f(x)| f(x) \in P_{3}, \operatorname{deg} f(x) = 3\}$ a subspace of $P_{3}$?

Given : $ U = \{f(x)| f(x) \in P_{3}, \operatorname{deg} f(x) = 3\}$ Does: U is a subspaces of $P_{3}$ I think the answer is yes. But in my textbook, they say no. And explain that zero is not in ...
2
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0answers
250 views

How can I find all the possible roots of a polynomial?

Is there any algorithm that can be used to find all the possible roots of a polynomial? For example, I'd like to find all possible roots of the polynomial $x^3 + 3x^2 + 2x + 6$. If I remember ...