Tagged Questions

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.

learn more… | top users | synonyms

1
vote
1answer
12 views

Lang problem about automorphism of polynomial ring

STATEMENT: Let $A$ be a commutative entire(integral domain) ring and $X$ a variable over $A$. Let $a,b\in A$ and assume that $a$ is a unit in $A$. Show that the map $X\mapsto aX+b$ induces a unique ...
0
votes
1answer
13 views

Deflating (factoring) a 6th degree polynomial

What is the procedure to factor a 6th degree polynomial of a complex variable? $$P(z)=1+x^2+x^3+x^4+x^5+x^6$$ I do have the correct answer but no idea how to get to it. The answer is: ...
0
votes
1answer
12 views

Limit of multivariate polynomial with large arguments

If I have a polynomial $f(x,y)=x^4+y^4-4xy$, how would I go about showing that as the standard norm of $(x,y)$ goes to infinity, $f(x,y)$ goes to infinity?
0
votes
0answers
42 views

The galois group of the polynomial $x^9+x^3+1$

What is the galois group of the polynomial $x^9+x^3+1$? Moreover give the bijection between subgroups and intermediate fields. Progress I think the order of the group is 108. But, there are many ...
0
votes
0answers
27 views

Points on a cubic curve

Regarding proof of a converse of a cubic equation's property: A curve passes through 3 collinear points (A, C and E) on x - axis. Points B and D are respectively midpoints of CA and CE. If the ...
4
votes
3answers
239 views

Finding the all roots of a polynomial by using Newton-Raphson method.

Is there a general formulation for finding all roots of a polynomial, especially the complex ones, by using the Newton-Raphson Method?
1
vote
2answers
24 views

Maximal ideal in $\mathbb{Q}[x,y]$

I am trying to prove that $(x,y)$ is a maximal ideal of $\mathbb{Q}[x,y]$. Since an ideal $I \subseteq R$ is maximal if and only if $R/I$ is a field, it suffices to prove that $\mathbb{Q}[x,y]/(x,y)$ ...
0
votes
1answer
17 views

Sequence of Polynomials and Weierstrass Approximation Theorem

Using the Weierstrass Approximation Theorem I need to prove that if $f$ has $k$ continuous derivatives in $[a,b]$, then exists a sequence of polynomials {$P_n$} such that {$P_n$} converges uniformly ...
0
votes
1answer
18 views

polynomial modulo polynomial

If $h(x) = x^2 + 1$, $g(x) = x^2 + x + 1$ and $f(x) = x^3 + x + 1$, then $$ \begin{align} g(x)h(x) \mod f(x) &\equiv (x^2 + x + 1)(x^2 +1) \mod x^3 + x + 1 \\ &\equiv x^4 + x^3 + 2x^2 + x + ...
0
votes
1answer
24 views

How many roots does a complex polynomial has?

Define $f(z)=z^4-4z^3+8z-2$. Find how many zeros (including multiplicity) the function has in $\{z\in\mathbb{C}:|z|<3\}$. I tried using Rouché's-theorem on $\{z\in\mathbb{C}:|z|<3\}$. The ...
2
votes
0answers
14 views

Help me find $S_{-2}$ using the polynomial

I found the equation , then I found $S_3$ : $$S_3-S_1-3=0$$ And I found $S_3$ Also I gave it a try using the Generalized form: $$S_n+S_{n-2}+S_{n-3}=0$$ Let $n=0$ $$S_1+S_{-1}+S_{-2}=0$$ Now ...
1
vote
1answer
9 views

Polynomial Factoring over a finite field

Ok, so I'm trying to figure out how to factor polynomials over a finite field. My polynomial is x^5 + x^2 + x + 1 and I have to factor over GF(2) I know the answer is (x+1)^2 * (x^3 + x + 1), because ...
1
vote
2answers
36 views

An equation for the third powers of the roots of a given quadradic polynomial

The roots of the equation $3x^2-4x+1=0$ are $\alpha$ and $\beta$. Find the equation with integer coefficients that has roots $\alpha^3$ and $\beta^3$. GIVEN SR: $\alpha + \beta = \frac43$ PR: ...
2
votes
1answer
37 views

Is $g(A)$ diagonalizable?

Let $A$ be an $n \times n$ diagonalizable matrix; let $g(x)$ be a polynomial. Is $g(A)$ diagonalizable? If not, what are the minimum hypothesis one needs to make so that it works (if any?) (As ...
1
vote
4answers
74 views

How to factor $x^{4}-22x^{2}+9$ over real numbers?

How do you factor $f(x) = x^{4}-22x^{2}+9$ over real numbers? I know that over integers it is $(x^2-4x-3)^2$.
0
votes
1answer
35 views

Need to factorize$ x^7 - x$ to irreducible polynomials over $\operatorname{GF}(2)$

Can some one help me with an easy method to solve this question? Factorize $x^7 - x$ to irreducible polynomials over $\operatorname{GF}(2)$
0
votes
0answers
17 views

For what values of indices is this polynomial divisible by $x^3-x^2+x-1$?

I need to find the values of the positive integer $n$ such that the polynomial: $$ f(x)=x^{3n}-x^{2n}+x^n-1 $$ is divisible by the polynomial: $g(x)=x^3-x^2+x-1$. I ...
2
votes
2answers
19 views

Remainders on division of polynomials

I am told that the remainder on division of a polynomial $p(z)$ by $z^3+z^2+z+1$ is $z^2-z+1$. I am also given that $p(1)=2$, and then asked to determine the remainder when $p(z)$ is divided by ...
3
votes
0answers
26 views

Condition $|x_1x_2+1|<x_1+x_2$ in quadratic polynomial

Let $x^2-ax+b$ be a polynomial with real coefficients having two nonzero roots. Given that $|b+1|<a$, and one of the roots have modulus $<1$, show that the other root has modulus $>1$. We ...
1
vote
2answers
59 views

Can we express the following ordinary generating function?

I wish to express the following power series $$ \sum_{k \ge 0} \binom{n-k}{m} x^k$$ where $n,m$ are positive integer such that $0< m \le n$
2
votes
3answers
50 views

Show that $\int_{-1}^1f(t)dt = \sum_{i=1}^nw_if(a_i)$

If $a_1,...,a_n$ are distinct reals, show that there are scalars $w_1,...,w_n$ such that $$\int_{-1}^1f(t)dt = \sum_{i=1}^nw_if(a_i)$$ for all polynomials $f(t)$ in $P_{n-1}$. I ...
0
votes
1answer
22 views

Finding constants to make f(g(x)) and g(f(x)) equal.

Let $f(x) = ax + b$ and $g(x) = cx^2$, where $a$, $b$, and $c$ are constants. Compute $f\circ g$ and $g\circ f$. Determine for which constants $a$, $b$, and $c$ it is true that $f\circ g =g\circ f$. ...
0
votes
0answers
15 views

finding max and min value of polynomial function with starting point and steps

My instructor said you will find max and min value of a polynomial function. However, you will use a point that is called starting of search and step size. I don't understand starting of search and ...
5
votes
1answer
32 views

Quadratic polynomial with alternate negative value

Let $f(x)=-x^2+ax+b$, where $a,b\in\mathbb{R}$. Suppose there exist distinct integers $m,n$ such that $f(m)=-n^2$ and $f(n)=-m^2$. Prove that there are infinitely many pairs of integers $x,y$ such ...
1
vote
0answers
21 views

generalization of geometric series $ \sum_{k=0}^\infty x^{p(k)} $

I have been playing around with infinite series and I wondered if it is possible to find an expression for the series: $$ \sum_{k=0}^\infty x^{p(k)} $$ as a generalization of geometric series. $p(k)$ ...
0
votes
1answer
42 views

Why are these two quotients equal?

I'm not being able to check why are these two quotients equal. $\mathbb C[x]/(x^2-x^3)= \mathbb C[x]/(x^2)$ Can someone tell me why is it valid?
2
votes
2answers
49 views

How to find a minimal polynomial

I need to find minimal polynomial of $\alpha = \sqrt 2 + \sqrt [3] 3 $ over $\mathbb Q$ and prove that my result is minimal polynomial. How do I do that?
3
votes
0answers
28 views
+100

Polynomial factorisation - absolute value of coefficients

This question takes the factorisation of a polynomial $p(x)=q(x)r(x)$, where $p$ (and for my purpose here $q$ and $r$) have integer coefficients and asks if the maximum absolute value of the ...
3
votes
1answer
91 views

Maximum absolute value of polynomial coefficients

Suppose we have a polynomial in integer coefficients $$p = p_0 + p_1 x + p_2 x^2 + \ldots + p_n x^n, p_k \in \mathbb{Z}$$ Now define $M(p)$ as the maximum absolute value of the coefficients of $p$, ...
1
vote
2answers
25 views

How to find an alternate form of this polynomial (factorize?)

I am trying to find the limit of the function $$\lim_{t \to 1} {{t^3-2t+1}\over{t^3+t^2-2}}$$ And it obviously evaluates to ${0\over0}$ so at first glance it is indetermined. But I have these two ...
1
vote
1answer
48 views

Linear Differential Operator Property

One of my exercises asks the following. Let $D\colon\mathbb R[X]\to\mathbb R[X]$ (where $\mathbb R[x]$ is the space of polynomials with real coeffients) be the differential operator ...
0
votes
2answers
50 views

Finding roots of $x^9 + 1$ modulo $19$

As part of a problem to factorise $f = x^6 + x^3 + 1$ over $\mathbb F_{19}$, I've realised that $f$ is a factor of $x^{18} - 1 = (x^9 + 1)(x-1)(x^6 + x^3 + 1)(x^2 + x + 1)$ which splits into linear ...
2
votes
2answers
37 views

Relation among numbers in a triangle like tartaglia:

I have a polynomial associate to a number: $$\begin{align} k&=1 &n-1\\ k&=2 &n^2-2n +2\\ k&=3 &n^3-3n^2+6n+6\\ k&=4 &n^4-4n^3+12n^2-24n+24 \end{align}$$ and in ...
1
vote
0answers
32 views

monic irreducible polynomial over K

If $F/K$ is a field extension, $X^n-a \in K[x]$ is irreducible, $\alpha \in F$ is a root and $m \in \Bbb N$ a divisor of $n$, prove that degree of $\alpha^m$ on $K$ is $\frac n m$. What is the monic ...
0
votes
1answer
36 views

Simplify using either Method 1 or 2. [on hold]

I need to simplify $$\dfrac{1 - \dfrac 4{t+5}}{\dfrac{4}{t^2-25} + \dfrac{t}{t-5}}$$ I am posting the methods I can use as images. !
0
votes
1answer
16 views

Converse of the implication $V(S)\subseteq V(T)\iff T\subseteq\sqrt{\langle S\rangle}$.

I'm having trouble recalling one direction of the following bi-implication. Suppose $S,T$ are subsets of the polynomial ring $k[X_1,\dots,X_n]$ over an algebraically closed field. We have ...
7
votes
1answer
51 views

Inequality relating coefficients and roots of a complex polynomial

While going through some olympiad handouts I stumbled upon a problem related to an upper bound for the Mahler measure, which stated that Given a polynomial $f(x) = x^n + a_{n-1}x^{n-1} + \dots + a_0 ...
1
vote
0answers
20 views

Polynomial representation of intersection of polynomials

How to minimally represent intersection of two degree $d$ polynomials intersecting at $d^2$ points as a single polynomial?
0
votes
0answers
29 views

What is it called when we interpolate a point INTO a grid…

Consider a uniform 2D grid, where each $(x,y)$ value on this grid has a corresponding value. So, if I want to find the value, $v$ (unknown) of a point that exists at some arbitrary co-ordinate $(x,y)$ ...
0
votes
2answers
43 views

Please solve these polynomials [closed]

Write in lowest terms: First: $r^3-3r^2+2r-6$ divided by $21-7r$ i.e. $$\frac {r^3-3r^2+2r-6}{21-7r}$$ Second: Multiply $36-w^2$ divided by $wt+6t-w-6$ by $8t-8$ divided by $2w^2-11w-6$ i.e. ...
0
votes
2answers
46 views

Sequences of polynomial functions converging uniformly on $[a,b]$ to a continuous function not a polynomial

What is (are) the necessary and sufficient condition(s), if any, for a sequence of polynomial functions to converge uniformly on a given (finite) closed interval $[a,b]$ to a continuous function not a ...
2
votes
4answers
56 views

Can any of these polynomials be a square?

It's in $\mathbb C[x]$ and they have the form $$1+x+x^2+\dots +x^n$$ Obviously $n$ can't be odd. I can prove it for any specific polynomial via GCD with the derivative, but how to prove it for all $n$ ...
2
votes
2answers
34 views

Extension field, degree of $[\mathbb Q(i,\sqrt{-3}):\mathbb Q]$

I want to calculate the degree of $[\mathbb Q(i,\sqrt{-3}):\mathbb Q]$, can I do like that: $$X=i+\sqrt{-3}\implies X=i(1+\sqrt{3})\implies X^2=-(1+\sqrt{3})^2\implies X^2=-1-2\sqrt{3}-3\implies ...
0
votes
1answer
31 views

Can we prove the Riemann-Lebesgue lemma by using the Weierstrass approximation theorem?

I'd like to prove the following version of the Riemann-Lebesgue lemma: Let $f: [0,1] \to \mathbb R$ be continuous. Then $$\int_0^1 f(x)\sin(nx) \, dx \xrightarrow{n \to \infty} 0$$ It's quite ...
1
vote
1answer
30 views

Commutative Diagrams and Polynomials

Recently, considering how algebraic numbers may be defined by the algebraic properties that they satisfy (and in particular, the polynomials of which they are roots), I started to wonder about, for ...
0
votes
0answers
24 views

Writing solution to an arbitrary ODE with arbitrary initial values as the sum of a power series?

Let $f(t), g(t)$ be polynomials, and let $y$ be a function of $t$. Given the ODE $y'' + f(t) y' + g(t) y = 0$ with initial conditions $y(0) = \alpha$ and $y'(0) = \beta$, write $y$ as the sum of a ...
0
votes
3answers
59 views

Show $x^3-3x^2-3x+7$ has a positive real root.

How to show that the polynomial $$x^3-3x^2-3x+7$$ has a positive real root? I can graph it and see that it is indeed true, but can we prove it rigorously?
0
votes
0answers
11 views

multivariate polynomials with common roots

Let $f$ and $g$ be two multivariate polynomials that vanish on certain common points over $\Bbb R^n$. Can we tell anything about the structure of $f$ and $g$? In the univariate case, we have gcd is ...
1
vote
2answers
22 views

Prove that the number of automorphisms in $\mathbb Q[\alpha]$ equals $1$ $(|Aut\mathbb Q[\alpha]|)=1$

Please, help me to understand this problem: Let $\alpha=\sqrt[3]{2}$ be a root of the polynomial $x^3-2$. a) Prove that the number of automorphisms in $\mathbb Q[\alpha]$ equals $1$ $(|Aut\mathbb ...
7
votes
2answers
115 views

Solve this tough fifth degree equation.

$$x^5+x^4-12x^3-21x^2+x+5=0$$ I think it can be solved by trigonometric ways but how?