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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5
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4answers
253 views

If $f(x)$ is irreducible in $k[x]$, why is it also irreducible in $k(t)[x]$, for $t$ an indeterminate?

I've been thinking on this a few days, but I'm stuck. Let $k$ be a field, $f(x)$ irreducible in $k[x]$. Why is $f(x)$ also irreducible in $k(t)[x]$, for $t$ an indeterminate? I write ...
12
votes
5answers
3k views

Is there a simple explanation why degree 5 polynomials (and up) are unsolvable?

We can solve (get some kind of answer) equations like: $$ ax^2 + bx + c=0$$ $$ax^3 + bx^2 + cx + d=0$$ $$ax^4 + bx^3 + cx^2 + dx + e=0$$ But why is there no formula for an equation like $$ax^5 + ...
-3
votes
1answer
111 views

Showing $f(x)=g(x^p)$ in prime characteristic

Let $K$ be a field of characteristic $p \neq 0$ and let $f(x) \in K[x]$ be an irreducible polynomial with multiple roots inside some algebraic closure of $K$. Is it true that there exists $g(x) \in ...
1
vote
3answers
144 views

Given a system of quadratic equations $x^2-a_ix+b_i=0$, can all of the coefficients $a_i$, $b_j$ be solution to one of these above equation?

Here are $n$ quadratic equations ($n>1$): $$x^2-a_ix+b_i=0\quad(i=1,\ldots, n)$$ where the $a_i$, $b_i$ are distinct. Can all of the $a_i,b_i$ be roots of one of the above equations?
3
votes
5answers
242 views

Proving Quadratic Formula

purplemath.com explains the quadratic formula. I don't understand the third row in the "Derive the Quadratic Formula by solving $ax^2 + bx + c = 0$." section. How does $\dfrac{b}{2a}$ become ...
1
vote
1answer
113 views

minimal polynomial of normal endomorphism with given eigenvalues

What's the minimal polynomial of a normal endomorphism $\phi$ with eigenvalues $2, 2, 1+i, 1+i, 1-i, 1-i, 3$? It is $\mu_\phi | (t-2)^2(t-1-i)^2(t-1+i)^2(t-3)$ but is there any more I can derive from ...
1
vote
3answers
101 views

How to show that $h(x^p) \equiv h(x)^p \pmod{p}$? [duplicate]

Possible Duplicate: Why $g(x^{p})=(g(x))^{p}$ in the reduction mod $p$? Let $h(x) \in \mathbb{Z}[x]$ and $p$ be a prime. We know that for any integer $\alpha$ we have that $\alpha^p \equiv ...
3
votes
3answers
122 views

Proving $\frac{-\theta + \theta^2}{2}$ is an algebraic integer in $K = \mathbb{Q}(\theta)$, given that $\theta^3 + 11\theta - 4 = 0$

As the title says, given that $\theta^3 + 11\theta - 4 = 0$, I'm trying to prove that $\frac{-\theta + \theta^2}{2}$ is an algebraic integer in $K = \mathbb{Q}(\theta)$. I know that $x^3 + 11x -4$ ...
1
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1answer
129 views

Intersection of a line with a curve

I have the following question: Given a line $y=\theta(1-x)$ where $0<x<1$, $0<y<1$ and $0<\theta<1$, I have a collection of curves $$ y^K=1-(1-x)^K $$ parametrized by an ...
4
votes
1answer
1k views

What are the best methods for solving cubic and quartic equations by computer programs?

We know that there are closed form formulas for real roots of degree 4 and 3 polynomials, but people sometimes advise to use numerical (e.g. Newton) methods anyway. They claim that closed formulas ...
4
votes
6answers
313 views

How to prove that $\frac{10^{\frac{2}{3}}-1}{\sqrt{-3}}$ is an algebraic integer

As the title says, I'm trying to show that $\frac{10^{\frac{2}{3}}-1}{\sqrt{-3}}$ is an algebraic integer. I suppose there's probably some heavy duty classification theorems that give one line ...
6
votes
6answers
1k views

Irreducibility of $x^5 -x -1$ by reduction mod 5

Is there a quick way of deducing that $x^5-x-1 \in \mathbb{Z}[x]$ is irreducible by reducing it mod 5, other than verifying that it has no roots in $\mathbb{Z}_5$ and no factorization as the product ...
3
votes
2answers
351 views

How to find all polynomials $P(x)$ with real coefficents satisfying $P^2(x)-1=4P(x^2-4x+1)$

Find all polynomials $P(x)$ with real coefficents satisfying $P^2(x)-1=4P(x^2-4x+1)$. My solution: Let the first term of $P(x)$ be $ax^n$. We see first term of left side is easily $a^2x^{2n}$ ...
6
votes
1answer
744 views

Zeros of a complex polynomial

The question is: Show that $$ P(z) = z^4 + 2z^3 + 3z^2 + z +2$$ has exactly one root in each quadrant of the complex plane. My initial thought was to use Rouche's Theorem (since that's generally ...
3
votes
1answer
117 views

How to solve this System of Polynomial Equations?

I have to complete a summer packet of 90 Algebra 2 questions. I have completed 89 of them, the only one I could not get was this. I know the answer is $y = \frac {47}2$, $\frac 17$ according to ...
1
vote
3answers
752 views

What's polynomial composition useful for?

I've made this question here. But I got no answer then I'll make a question with it: I remember of studying: Sum of two polynomials; Difference of two polynomials; Product of a constant and a ...
1
vote
0answers
368 views

Fitting a 3d point cloud with a polynomial surface

I have 3D point cloud and I would like to fit a polynomial surface to it. Could anybody please explain the step by step process to that. Thanks a lot.
2
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1answer
301 views

Composition of two polynomials

How's to make the composition of two polynomials? According to this page: If $ P = (x^3 + x) $, $ Q = (x^2 + 1) $ then, $ P\circ Q = P\circ (x^2 + 1) = (x^2 + 1)^3 + (x^2 + 1) = x^6 + 3 x^4 + 4 x^2 ...
1
vote
1answer
69 views

Understanding a theorem of Marden's on the moduli of zeros of polynomials

My question is concerning Theorem 3.2 in this paper of Marden's. The gist of the theorem is stated below. Theorem 3.2. Every polynomial of the form $$ f(z) = \sum_{j=0}^{n} (b_j - ...
2
votes
1answer
135 views

Farey sequences for polynomials?

Does a notion of Farey sequence (or something equivalent) exist for polynomials over finite fields?
2
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1answer
170 views

Polynomial ring over $\mathbb{Z}_2$

As $f(x)$ is an irreducible over $\mathbb{Z}_2[x]$ so $R/(f)$ is an infinite field. Am I right?
2
votes
3answers
526 views

If $f$ is a polynomial of degree $n$, then $f(x) \equiv 0\pmod p$ has at most $n$ solutions.

I know that I have to prove this by induction on $n$ when we let $f(x)=a_{n}x^n + a_{n-1}x^{n-1} +\cdots + a_0$. I have two books in front of me with the complete proof but I don't see how after they ...
1
vote
1answer
111 views

Question about the degree of the inverse of a polynomial $p(x)$ modulo $q(x)$

I have a doubt about the modular inverse polynomial degree. Let $p(X)$ be a polynomial in the ring $F[X]$ with $\deg(p)=\delta$, where $F$ is a finite field with characteristic 2. If ...
1
vote
1answer
666 views

Solving a polynomial modulo an integer

Say I have a polynomial $F$ of degree $n$ with coefficients in $Z_m$ and I wish to find $x$ such that $F(x)=0$ (mod $m$). For instance if $F(x)=x^{2}-a$ the solution would be the modulo $m$ squareroot ...
0
votes
1answer
302 views

Generating a random monotonically increasing polynomial?

Given a polynomial $y : \mathbb{R} \mapsto \mathbb{R}$ of degree $p$: $$ y(x) = \sum_{k=0}^p c_k\, x^k,$$ can a random set of coefficients $\{c_0, \cdots ,c_p\}$ be generated such that $y$ is ...
1
vote
1answer
154 views

For which values of $n$ is the polynomial $p(x)=1+x+x^2+\cdots+x^n$ irreducible over $\mathbb{F}_2[x]$?

For which values of $n$ is the polynomial $p(x)=1+x+x^2+\cdots+x^n$ irreducible over $\mathbb{F}_2[x]$ ? E.g. $x+1$, $x^2+x+1$ are irreducibles. Subcase of this question Factor by irreducible ...
4
votes
2answers
239 views

For what $(n,k)$ there exists a polynomial $p(x) \in F_2[x]$ s.t. $\deg(p)=k$ and $p$ divides $x^n-1$?

For what $(n,k)$ there exists a polynomial $p(x) \in F_2[x]$ s.t. $\deg(p)=k$ and $p$ divides $x^n-1$? Motivation: if exists $p(x)$, then ideal generated by $p(x)$ is "cyclic error correcting code". ...
3
votes
4answers
410 views

What will be the value of $P(12)+P(-8)$ if $P(x)=x^{4}+ax^{3}+bx^{2}+cx+d$?

What will be the value of $P(12)+P(-8)$ if $P(x)=x^{4}+ax^{3}+bx^{2}+cx+d$ provided that $$P(1)=10$$ $$P(2)=20$$ $$P(3)=30$$ I put these values and got three simultaneous equations in $a, b, c, d$. ...
0
votes
1answer
312 views

Count all degree 2 monic irreducible and not irreducible polynomials

There's this exercise that really has kept me stuck for a day by now, will you please help me figure out: let's consider polynomials in $\mathbb Z_3$: characterize degree 2 not irreducible monic ...
2
votes
3answers
210 views

How to find $P(n+1)$, given $P(x)$ for $x = 0,1,\ldots,n$?

Given $P$, a polynomial of degree $n$, such that its values at $n+1$ points are $P(x) = r^x$ for $x = 0,1, \ldots, n$ and some real number $r$, I need to calculate $P(n+1)$? Can this be done without ...
5
votes
2answers
203 views

General solution of $C_{n+2}(x)=xC_n(x)+nC_{n-1}(x)$

Airy differential equation. $y''(x)=xy(x)$ $y'''(x)=y(x)+x y'(x)$ $y'^v(x)=x^2y(x)+2 y'(x)$ $y^v(x)=4xy(x)+x^2 y'(x)$ $y^{(6)}(x)=(x^3+4)y(x)+6x y'(x)$ . . $y^{(n)}(x)=A_n(x)y(x)+B_n(x) y'(x)$ ...
5
votes
1answer
186 views

A Gröbner Basis Computation Gone Bad

Here is the problem statement: Consider the polynomial ideal $I = \langle b-r_1-r_2, c-r_1r_2 \rangle \subset \mathbb{Q}[r_1,r_2,b,c].$ Show that $I \cap \mathbb{Q}[b,c] = \langle 0 \rangle$. ...
3
votes
1answer
474 views

Multiple choice question - number of real roots of $x^6 − 5x^4 + 16x^2 − 72x + 9$

The equation $x^6 − 5x^4 + 16x^2 − 72x + 9 = 0$ has (A) exactly two distinct real roots (B) exactly three distinct real roots (C) exactly four distinct real roots (D) six distinct real roots
6
votes
4answers
1k views

Find all real solutions to $8x^3+27=0$

Find all real solutions to $8x^3+27=0$ $(a-b)^3=a^3-b^3=(a-b)(a^2+ab+b^2)$ $$(2x)^3-(-3)^3$$ $$(2x-(-3))\cdot ((2x)^2+(2x(-3))+(-3)^2)$$ $$(2x+3)(4x^2-6x+9)$$ Now, to find solutions you must set ...
2
votes
5answers
437 views

Factor $4x^3-8x^2-25x+50$ completely

Factor $4x^3-8x^2-25x+50$ completely The highest numbers you can take would be $1$, $2$, or $4$. Neither of those apply to all. So let's try the $x$! But the last term $50$ doesn't have an $x$ ...
6
votes
2answers
349 views

If two polynomials are equal as functions, are they necessarily equal as polynomials?

Say you have a finite field $F$ of order $p^k$. Suppose that $f,g\in F[X_1,\dots,X_m]$, such that the degree of each $X_i$ is strictly less than $p^k$ in both $f$ and $g$. I'm putting this condition ...
2
votes
1answer
124 views

If the polynomial $f$ is zero on the nonzero set of another polynomial $g$, does $f=0$?

Suppose $f(x_1,\dots,x_n)$ is a polynomial in $n$ indeterminates over an infinite field $F$. Suppose $f((a_i))=0$ for all $n$-tuples $(a_i)$ such that $g((a_i))\neq 0$, where $g(x_1,\dots,x_n)$ is ...
6
votes
1answer
196 views

Is there a name for these polynomials?

Given $t \in \mathbb{R}[0,1]$, consider the following set of polynomials: $$ \left[-{\left(t - 1\right)}^{2} t, {\left(t - 1\right)} {\left(t^{2} - t - 1\right)}, -{\left(t^{2} - t - ...
5
votes
2answers
256 views

$F, G \in k[X_1, \dots , X_n]$ homogeneous of degrees $r$ and $r+1$ $\implies$ $F+G$ is irreducible

I have a question about Exercise 2-34 from William Fulton's Algebraic Curves book. The exercise is as follows. Suppose that $F, G \in k[X_1, \dots , X_n]$ are forms (i.e. homogeneous ...
10
votes
1answer
2k views

Why does synthetic division work?

Synthethic division is commonly taught, but I have never actually had a proof/explanation shown to me. Why does it work? Work So Far I related the "$x$" to powers to 10, and then proceeded to relate ...
-1
votes
1answer
141 views

Find a degree 5 polynomial $f \in \mathbb Z_5[x]$ with exactly 4 distinct roots

Find a degree 5 polynomial $f \in \mathbb Z_5[x]$ so that it has exactly 4 distinct roots and factorize it as product of irreducible factors. I'm really struggling in finding such polynomial, so ...
1
vote
1answer
106 views

Factorize $f$ as product of irreducible factors in $\mathbb Z_5$

Let $f = 3x^3+2x^2+2x+3$, factorize $f$ as product of irreducible factors in $\mathbb Z_5$. First thing I've used the polynomial reminder theorem so to make the first factorization: ...
2
votes
3answers
128 views

question related to radical sign

My question is- Let $p(x)= \sqrt{x + 2 + 3\sqrt{2x-5}} - \sqrt{x - 2 + \sqrt{2x-5}}$. Then $p(2012)^6$ equals? Any solution for this question would be greatly appreciated. Thank you,
0
votes
2answers
73 views

Find monic grade 3 polynomial in $\mathbb Z_p[x]$ then factorize

Let $f = 15x^4+22x^3-x=0$ a polynomial in $\mathbb Z_p[x]$, find the first prime $p$ value that will make $f$ result in being grade 3 and monic. Then factorize $f$ in $\mathbb Z_3[x]$ as product of ...
0
votes
1answer
65 views

How to simplify $(3a-b^2-a)^3$ by using special product?

When I simplify $(3a-b^2-a)^3$, I used $(u±v)^3=u^3±3u^2v+3uv^2±v^3$ but I'm confused with $(b^2-a)$
2
votes
2answers
61 views

Sum of the thirteenth power of the roots of given polynomial

Find the sum of the thirteenth powers of the roots of $x^{13} + x - 2\geq 0$. Any solution for this question would be greatly appreciated.
2
votes
3answers
110 views

Theorem for Dividing Polynomials

When a polynomial $$P(x)=x^4- 6x^3 +16x^2 -25x + 10$$ is divided by another polynomial $$Q(x)=x^2 - 2x +k,$$ then the remainder is $$x+a.$$ I have to find the values of $a$ and $k$. Can ...
4
votes
1answer
132 views

Is $(x^3-x^2+2x-1)$ prime in $\mathbb{Z}/(3)[x]$?

This is somewhat of a follow up on this question: Why is $(3,x^3-x^2+2x-1)$ not principal in $\mathbb{Z}[x]$? I'm curious, is $\mathbb{Z}[x]/I$ a domain, with $I=(3,x^3-x^2+2x-1)$? I know $I$ is not ...
0
votes
2answers
365 views

Why can/do we multiply all terms of a divisor with polynomial long division?

I'm trying to understand why polynomial long division works and I've hit a wall when trying to understand why we multiply all terms of the divisor by the partial quotient. Consider: $$\frac{x^2 + 3x ...
0
votes
2answers
83 views

Find a prime number $p$ so that $f = \overline{3}x^3+ \overline{2}x^2 - \overline{5}x + \overline{1}$ is divided by $x-\overline{2}$ in $\mathbb Z_p$

Let $f = \overline{3}x^3+ \overline{2}x^2 - \overline{5}x + \overline{1}$ be defined in $\mathbb Z_p$. Find a prime number $p$ so that $f$ can be divided by $g = x-\overline{2}$, then factorize $f$ as ...