This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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3
votes
1answer
25 views

What causes the equating-the-coefficients method not to work?

When searching here to find methods for solving quartic polynomials, I came across a question where one of the solutions (at the very end) mentions that the equating coefficients method can fail. ...
-1
votes
2answers
43 views

Finding the range of the function $\frac{3}{f(x)}$ [on hold]

Given that ${f(x)=2x^2+4x+3}$, find the range of the function $\frac{3}{f(x)}$.
0
votes
1answer
22 views

Algorithm to find the irreducible polynomial

What algorithm can be used find an irreducible polynomial of degree $n$ over the field $GF(p)$ for prime $p$. The reason I ask is I want to make a program for finite field arithmetic, but creating a ...
4
votes
0answers
40 views

if $x^k-x\in\mathbb{Z}$ and $x^l-x\in\mathbb{Z}$, then $x\in\mathbb{Z}$?

is it true that for any $k,l\in\{2,3,4,\dots\}$, $k\neq l$, if $x\in\mathbb{R}$ satisfies $x^k-x\in\mathbb{Z}$ and $x^l-x\in\mathbb{Z}$, then $x\in\mathbb{Z}$? This is a generalisation of if ...
4
votes
2answers
32 views

Let $P$ be a 4-th degree real polynomial with 5 conditions given. How to compute $P(4)$?

Yesterday I was math tutoring a 18-years old girl. And she asked me for the following problem: given $P\in\Bbb R[X]_4$, i.e. $P$ a real polynomial of degree exactly $4$, such that: $P(1)=0$ It has a ...
6
votes
4answers
295 views

Finding coefficient of polynomial?

The coefficient of $x^{12}$ in $(x^3 + x^4 + x^5 + x^6 + …)^3$ is_______? Somewhere it explain as: The expression can be re-written as: $(x^3 (1+ x + x^2 + x^3 + …))^3=x^9(1+(x+x^2+x^3))^3$ ...
2
votes
1answer
32 views

In general, when does a ring have a division algorithm?

I'm working through Herstein's "Abstract Algebra" text, and am currently working through section 5. Theorem 4.5.5 introduces the division algorithm for polynomial rings over fields, which states: ...
2
votes
1answer
29 views

Showing that $P(x)=x^{p-1}-1+pQ(x)$.

This comes from a problem from Imo math notes on algebraic extensions. One needs to show that $P(x)=(x+1)(x+2) \dots (x+p-1) = P(x)=x^{p-1}-1+pQ(x)$, where $p$ is prime and $Q(x)$ a polynomial with ...
-6
votes
1answer
40 views

A question on polunomial [on hold]

Let $m\in (0,1)$ and ${a_n}{x^n} + .... + {a_1}{x^1} - f(m) = 0$ and $x\in \mathbb{C}$ $f(m) $ is continuous decreasing function of $m$. $a_i\ge0$ for all $i$. $k(m)$ is positive zero of first ...
0
votes
0answers
27 views

About two polynomials $f,g$ such that $f=\pm g$

Let $R$ be an infinite commutative ring with unit and with characteristic zero. Assume that $f,g\in R[x_1,...,x_n] $ are nonzero and such that $f(x_1,...,x_n)=s(x_1,...,x_n) g(x_1,...,x_n)$, where ...
0
votes
0answers
19 views

Is Horner Algorithm works or suitable for polynomial with complex coefficients?

I have checked Algorithm of Horner for polynomial factorization with complex coeffecients ,I got it works but i can't show it works in general for any polynomial with complex coeffeciant . My ...
0
votes
0answers
22 views

$f\in \mathbb{Z}[x], f(x) = y^2, f(y) = z^2, f(z)=x^2 \implies x=y=z$?

Given that $f\in \mathbb{Z}[x], f(x) = y^2, f(y) = z^2, f(z)=x^2$ for some real numbers $(x,y,z)$, does it follow that $x=y=z$? It is well known that if $f\in \mathbb{Z}[x]$ and $f(x)=y, f(y)=z, ...
0
votes
0answers
9 views

Small roots of multivariate polynomials

I'm looking for a program/algorithm (or even "theory"?) which checks if a given multivariate polynomial, say $P(x_1, \dots, x_n)$ has a \textit{real} root in some given region, say a closed ...
0
votes
2answers
51 views

Prove Expression cannot be factored

I am currently working on primes which can be expressed in form of a polynomial. For eg, Find all primes which can be expressed in form $n^4-52n^2+595$ It is very essential to tell whether a ...
1
vote
1answer
23 views

Show that the set of all polynomials $f$ in $F[x]$ such that $f(A)=0$ is an ideal

Let $A$ be an $n \times n$ matrix over a field $F$. Show that the set of all polynomials $f$ in $F[x]$ such that $f(A)=0$ is an ideal. I don't understand how to apply this when it comes to ...
0
votes
2answers
25 views

For complex polynomials $\gcd(f,g)=1$ if and only if $f$ and $g$ have no common root [on hold]

Assuming the fundamental theorem of algebra, prove the following. If $f$ and $g$ are polynomials over the field of complex numbers, then $\gcd(f,g)=1$ if and only if $f$ and $g$ have no common root.
0
votes
1answer
8 views

Zariski closure of the subset of a line.

Let $K$ be any field, and $F=V(f)$ a Zariski closed subset of $K^n$ such that $F\cap L$ is infinite for a given line $L$. Why does $L\subset F$ hold ? If $n=2$ for instance, we may assume that the ...
0
votes
0answers
48 views

Are polynomial roots special? [on hold]

Most functions have roots, and relations can also have roots (including complex roots) as well. They are where $f(x)=0$ for some $x$. (I'm considering only functions that actually will have roots.) ...
0
votes
0answers
34 views

How to find quadratic residues in the polynomial ring $k[t]$?

I have a question: Given a field $k$, finite or infinite, and an element $p(t)$ in the polynomial ring $k[t]$. I am searching for results of any kind about how to find quadratic residues in the ...
1
vote
3answers
75 views

If $a$ is a real root of $x^5 − x^3 + x − 2 = 0$, show that $\lfloor a^6 \rfloor = 3$.

If $a$ is a real root of $x^5 − x^3 + x − 2 = 0$, show that $\lfloor a^6 \rfloor = 3$. Obviously since this is a 5th degree polynomial, solving it is not going to be possible (or may be hard). ...
0
votes
1answer
17 views

If two monic polynomials have no common roots, are the coefficients of their product locally diffeomorphic to the product of the coefficients?

Let $P^d (t,\lambda)$ be the "generic" d-th degree monic polynomial $P^d (t,\lambda) = t^d + \sum\limits_{i=1}^d \lambda_i t^{d-i}$ with real coefficients. Let $\lambda(\xi,\eta)$ be given by the ...
1
vote
0answers
20 views

Prime and Maximal Ideals of $\mathbb{Z}[x]$ [duplicate]

Consider $R=\mathbb{Z}[x]$. Also let $p$ be a prime. Then we want to find all the prime and maximal ideals of $\mathbb{Z}[x]$. The prime ideals are $(0), (p), (x)$ and $(ap + bx)$. Then we see that ...
0
votes
2answers
26 views

how to find out the following statements are true or false?

Let $p(x)$ be an odd degree polynomial and let $q(x)=(p(x))^2+ 2p(x)-2$ a) The equation $q(x)=p(x)$ admits atleast two distinct real solutions. b) The equation $q(x)=0$ admits atleast two distinct ...
0
votes
1answer
47 views

how to prove whether statement is true or false?

Let $p(x)= x^n+\sum\limits_{k=0}^{n-1}a_k x^k$ and $q(x)= x^n+\sum\limits_{k=0}^{n-1}b_k x^k$ be two polynomials with real coefficients such that $x=3$ is a common root of the equations $p(x)=0$ and ...
3
votes
1answer
28 views

Why is a discrete algebraic subset of $K^n$ finite?

Let $K$ be any field. If $A$ is the zero set of a polynomial $P\in K[X]$, then $A$ is finite. This follows from the fact that $K[X]$ is Euclidian, using commutativity of $K$. Now let $A\subset K^n$ ...
0
votes
1answer
25 views

How polynomials are represented in matrix form for Univariate Polynomial. [on hold]

Represent this polynomial equation in matrix form $$P(x)=a_2 x^{2} +a_1x^{1} +a_0$$ ?
1
vote
1answer
47 views

A subset of a polynomial ring and its ideal. [duplicate]

Let $P=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$ and $I = (f)$ be a principal ideal in $P$ generated by $f \in P - \{0 \}$. Moreover let $L \subset \{x_1, \dots, x_n \}$ and $\hat{P} ...
1
vote
1answer
51 views

Find the greatest common divisor of pairs of polynomials

I'm trying to find the greatest common divisor of $$p(x)=7x^3+6x^2-8x+4$$ and $$q(x)=x^3+x-2$$ where both $p(x),q(x)\in\mathbb{Q}[x].$ And if $d(x)=gcd(p(x),q(x)),$ I need to find two polynomials ...
1
vote
0answers
31 views

$K$-affine $Hom$ functor relating to Polynomial Maps (Exercise in _Algebra: Chapter 0_ by Aluffi)

I'm currently learning some category theory and algebraic geometry, the basics at least. In Algebra: Chapter 0 by Aluffi, there is an exercise describing the relationship between morphisms of affine ...
1
vote
2answers
23 views

$f\in L[x,y]$ such that $f(x,0)=0$ implies $f=y g$ with $g\in L[x,y]$?

Suppose $L$ is an infinite field (or even algebraically closed; I'm not sure if it is necessary to add that hypothesis). If we have a polynomial $f(x,y)\in L[x,y]$ and $f(x,0)\equiv 0$, does that ...
1
vote
1answer
100 views

Determine all $k$ such that $k^3+k+1$ is divisible by 11

The task is the following: Determine all $\ k\in\mathbb Z$ such that $k^3+k+1$ is divisible by 11 I assumed that "$k^3+k+1$ is divisible by 11" is saying $11|k^3+k+1$. That means I can rewrite it as ...
3
votes
3answers
115 views

What is my special quadratic?

Start with $f(x)=x^2+bx+c$. Then, attempt to solve for $x$ in $f(x)=x$. It is easily found that $x=\frac{1-b\pm\sqrt{(b-1)^2-4c}}{2}$. Then, start again with $f(x)=x$ and apply the function $f$ to ...
3
votes
0answers
21 views

Is this a valid way for performing polynomial division?

While attempting to divide a quartic by a quadratic factor to find the other factors of the given quartic, I can't help feeling I "invented" a way of dividing polynomials. Suppose you have a quartic ...
4
votes
4answers
86 views

Real roots of the equation $\frac{18}{x^4} + \frac{1}{x^2} = 4$

I'm struggling a bit on the best method to find the real roots of the above equation. I ended up obtaining an equation of: $4x^4 - x^2 - 18 = 0$. Is this correct? From there on, how should I ...
1
vote
0answers
18 views

Does the resolvent cubic of the quartic equation always have at least 1 positive real root

I have written some code to solve for the roots of a 4th order polynomial, and in the process, I noticed that the resolvent cubic always has at least one positive real root. I can't find anywhere ...
4
votes
2answers
52 views

For which rings does a polynomial in $R$ have finitely many roots?

Which infinite rings satisfy the following Every non-zero polynomial in $R[X]$ has only finitely many roots ? Are there such rings which are not integral domains ?
0
votes
0answers
21 views

What is the probability of interpolating the Tutte polynomial of a planar graph from the values at the two hyperbolas?

Crossposted from MO The Tutte polynomial is a bivariate polynomial with positive integer coefficient which is a graph invariant and can be defined recursively. Evaluating it is $\#P$-complete even ...
1
vote
1answer
45 views

Checking whether a given polynomial is reducible or irreducible.

We're given the polynomial $x^{2}-2$ , and we need to prove that it's irreducible in $\mathbb Q$ but reducible in $\mathbb R$. Writing the polynomial as $(x^{2}-2) = 1.(x^{2}-2)$ , $(x^{2}-2)$ ...
1
vote
0answers
75 views

Solving a polynomial for cyclic roots

For any complex value of c, the following polynomial has 6 complex root values of p: $$1+c+2 c^2+c^3+p+2 c p+c^2 p+p^2+3 c p^2+3 c^2 p^2+p^3+2 c p^3+p^4+3 c p^4+p^5+p^6$$ For a general 6th order ...
2
votes
0answers
36 views

A subset of easily solved 4th degree polynomials

I've found (maybe, maybe not, but it's not on this Wikipedia or this Wikipedia) that there is a subset of easily solved quartic polynomials of the form ...
3
votes
1answer
53 views

Closed form for the sole positive root of the polynomial ${x^\alpha } + {x^{\alpha - 1}} + \cdots + {x^3} + {x^2} -p$, $p > 0$

For a study I'm making about the minimum and maximum radial values of bounded orbits in a central force system with general power law forces, I came across this special polynomial equation: ...
0
votes
0answers
19 views

Determining if a Polynomial is a subspace and its Basis

Hi, the question is Which of the subsets of P2 given in Exercises 1 through 5 are subspaces of P2 Find a basis for those that are subspaces. So I know that P'(1) = 1b + 2c And I know that P(2) = a ...
-6
votes
1answer
50 views

Polynomial Factoring problem [on hold]

Find the value of $p$ and $q$ such that $15x^3 + 26x^2 - 11x - 6$ is a factor of $15x^4 + px^3 - 37x^2 + qx + 6$.
0
votes
1answer
44 views

Finding a sixth degree polynomial that goes through 8 points

For a summative math research assignment, I will have to find a sixth degree polynomial that would ideally go through the following points: (0, 20.5625) (10, 27.5625) (30, 14.5625) (50, 14.6875) (60, ...
0
votes
2answers
36 views

How to prove if $5/2 < x < (5/4)(1+\sqrt2)$, then $25/(x(2x-5)\ge 8$

if $\frac52 < x < \frac54(1+\sqrt2)$, then $\frac{25}{x(2x-5)} \ge 8$ First I unpacked the conclusion to: $$ 16w^2-40w-25 \le 0 $$ I attempted to solve by manipulating the interval (squaring, ...
0
votes
0answers
32 views

property of complex polynomials

I can't solve the following problem: Let $p(z) = z^n + a_{n-1}z^{n-1} + ... + a_0$ be a complex polynomial of degree $n \ge 1$. Assume that there exist $j \in \{0, 1, ... n-1\}$ such that $a_j \neq ...
15
votes
2answers
152 views
+300

Decomposition of an algebraic number into a sum or product of algebraic numbers with smaller degree

An algebraic number can be identified by its minimal polynomial together with isolating intervals with rational bounds for its real and imaginary parts. The degree of an algebraic number is the degree ...
7
votes
1answer
143 views

Proving that one has solved chess by exhibiting the zeroes of polynomials over finite fields?

My question is based on one of Scott Aaronson blog post which states that a God-like being could convinced the villagers, to any degree of confidence, that she has solved chess by answering a few ...
5
votes
2answers
53 views

Roots of $x^p + x + [\alpha]_p \in \mathbb{Z}_p[x]$

Let $$g(x) = x^p + x + [\alpha]_p \in \mathbb{Z}_p[x],$$ where $p$ is prime. For which $\alpha, p \in \mathbb{Z}$ does $g(x)$ have at least one root? And for which $\alpha, p \in \mathbb{Z}$ ...
0
votes
1answer
53 views

Maximum of polynomial [on hold]

I was studying statics and came across this problem: Find the value $\beta$ such that $P$ has a maximum value in $R^2 - 1000^2 = P^2 + 2000P\cos(75^{\circ}+\beta)$. When $R$ is constant, the ...