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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7
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3answers
194 views

$X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$

I am trying to show that $X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$ where $n,m$ are positive integers. I showed that if $\gcd(n,m)$ is not $1$, then $X^n-Y^m$ is reducible. How to ...
3
votes
1answer
82 views

Prove that $(f(x)-x)^2 \not|f( f(…f(x)))) - x$

Let $f(x) \in \mathbb{R}[x], \deg f \geq 2$. Then $(f(x)-x)^2 \not|f( f(...f(x)))) - x$. I found this problem in my old notes, but there was no solution, and I could not remember one.
2
votes
0answers
68 views

Proof that $p(z)^2=a^2$ always has a nonreal solution.

Let $p(z)$ be a nonconstant integer polynomial of degree $n$ such that $p(0)=0$ and let $a$ be a nonzero real number. It seems that $$p(z)^2=a^2$$ Always has a nonreal solution (in $z$) if ...
1
vote
1answer
408 views

Algorithm to find the maximum/minimum of a polynomial without graphing.

For a quadratic equation of the form $y=ax^2+bx+c$ the max/min occurs at $x=-\frac{b}{2a}$. Is there any hard and fast equation like this for polynomials of degree $\geq 4$?. For such polynomials the ...
8
votes
5answers
301 views

Given $x+y$ and $x\cdot y$, what is $x^3+ y^3$ ?

I have been looking at an assortment of high school number sense tests and I noticed a reoccurring problem that states what x+y is and what $x\cdot y$ is then asks for $x^3+ y^3$. I want to know how ...
0
votes
1answer
99 views

Identifying some quotient rings

How come that $k[w,z]/(w^2+z,w^3 z^2)\cong k[w]/(w^7)$? Also why is $(xz,w)=(x,w)\cap(z,w)$ in the polynomial ring in 3 variables? what are the rules of ideal calculus making these results evident?
1
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2answers
109 views

Polynomials through successive differences

Let $h_0:\Bbb{N}\rightarrow\Bbb{N}$ be any function. Define recursively, for $m>0$, $$h_{r+1}(m)=h_r(m)-h_r(m-1).$$ Suppose that for some $k>0$ we have $h_k(m)\equiv d$ constant. Is this ...
0
votes
2answers
95 views

Polynomial such that $P(\sin x)=a\cos x+b$

Given four real numbers $a,b,\alpha,\beta$ with $ a\ne0, \alpha<\beta$. Does there exist a real coefficient polynomial $P(x)$ such that $$P(\sin x)=a\cos x+b$$ hold for all $x\in ...
1
vote
1answer
54 views

How to find charpoly from eigenvalues and CH to prove an equation

For an uknown 3x3 matrix $A$ we know that $\operatorname{tr} A = 0$, $\det(A) = 1/4$ and we also know that two eigenvalues are the same. Proove that $4A^3 = -3A - I$. Problem says to use Vieta to find ...
-3
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2answers
182 views

Sufficient and essential condition for polynomials $P$ and $Q$ to satisfy $P(\sin x)= Q(\cos x)$

The famous identity $\sin^2 x+\cos^2x =1$ can be written as follows: The polynomials $P(x)=x^2$ and $Q(x)=1-x^2$ satisfy $$P(\sin x)= Q(\cos x),\quad \text{for all }x\in\mathbb R$$ What are ...
1
vote
1answer
57 views

Factorize real polynomials to quadratic factors. Proof without fundamental theorem of algebra.

I've shown that if $P(x) \in \mathbb{R}[X]$, then exist $Q_1(X), \dotsc, Q_k(X) \in \mathbb{R}[X]$ so that $P(X) = Q_1(X) \cdots Q_k(X)$ with $\deg Q_i \leq 2$ for all $1 \leq i \leq n$. My proof - ...
2
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0answers
127 views

Rational Non-Integral Root

Prove by contradiction that the following equation with integral coefficients can not have a rational but non integral root. $x^{n}+p_{n-1}x^{n-1}+p_{n-2}x^{n-2}+\cdots +p_{0}=0$
4
votes
3answers
218 views

Solve $2000x^6+100x^5+10x^3+x-2=0$

One of the roots of the equation $2000x^6+100x^5+10x^3+x-2=0$ is of the form $\frac{m+\sqrt{n}}r$, where $m$ is a non-zero integer and $n$ and $r$ are relatively prime integers.Then the value of ...
2
votes
1answer
75 views

$P(-2)=P(-5)=n$

Prove that if $n$ is a positive integer, there exists only one polynomial $\displaystyle P(x)=\sum_{i=0}^n a_ix^i$ degree $n$ that satisfies: $(i):\,a_i\in\{0,1,\ldots,9\}$ $(ii):P(-2)=P(-5)=n$
1
vote
1answer
224 views

Subrings of polynomial rings over the complex plane

I have the following questions: (i) must every subring of the polynomial ring in two variables over the complex plane, containing the complex plane itself, be Noetherian? (ii) Are there Noetherian ...
1
vote
2answers
192 views

Show quartic polynomial has no real solutions

To show a lower bound for the runtime of an algorithm, I need to show that $$ 3 x^4 - \frac{64}{5} x^3 + \frac{192}{5} x^2 - \frac{192}{5} x+ 12 > 0 $$ for all real numbers $x\in \mathbb{R}$. ...
3
votes
1answer
83 views

Binomial coefficient difference

I have the following difference of binomial coefficients: $$f(m)={m+n\choose n}-{m-d+n\choose n}$$ I believe the following two things should hold true: For $m$ large enough, $f(m)$ is a polynomial ...
1
vote
2answers
59 views

Is this a theorem regarding the solutions of polynomials?

I wanted to refer to this, but I can't remember if this a theorem, named or otherwise, and if it is, how to properly state it. The idea is if we have a solution in radicals to a polynomial with ...
1
vote
1answer
254 views

Galois group of a quartic

Let $x^4+ax^2+b$ in $K[x]$ (with char $K\neq $2) be irreducible with Galois group $G$. (a) If $b$ is a square in $K$, then $G = \mathbb{Z}_2\times\mathbb{Z}_2$. (b) If $b$ is not a square in $K$ ...
0
votes
1answer
113 views

Is this polynomial solvable by radicals?

The polynomial $p(x) = x^6-9x^4-4x^3+27x^2-36x-23$. has at least one (real, irrational) root that is expressible by radicals (can you find it?). Are all the roots of $p$ expressible by radicals and ...
0
votes
1answer
87 views

Substitution to linear + nth power form

Given an arbitrary polynomial: $$a_0 + a_1x + a_2x^2 ... a_nx^n$$ Does there exist a series of substitutions (or single substitution if you choose to combine them) that leaves this function in the ...
5
votes
3answers
3k views

Polynomial of degree 4 with real coefficients, two complex roots given.m

Write in the form f(z) = 0, where f(z) is a polynomial of degree 4 with real coefficients, the equation having (3 + i) and (1 + 3i) as two of its roots. Can anyone help me? I'm guessing the two ...
2
votes
1answer
128 views

How do I determine between positive and negative inflection

Is it possible to identify whether an inflection point such as this example, contained in y = x^3 from the wikipedia: Is positive or negatively oriented (i.e. the ...
1
vote
2answers
95 views

Why $S_4$ has no transitive subgroups of order 6?

I know that every transitive subgroup of $S_4$ have to be order divisible by 4, but i should solve this with Galois Theory. I think this theorem can be usefull: Theorem 4.2. Let K be afield and f in ...
0
votes
3answers
79 views

How do I see that $x^5+x-1=(x^2-x+1)(x^3+x^2-1)$

I've recently been asked my friend to find the solutions to the expression $x^5+x=1$, now I haven't yet done complex analysis, but I thought I'd give it a go. I came up with a pretty, but probably ...
1
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2answers
134 views

Finding a least common multiple (LCM)

My Algebra 2 book explains how to find a least common multiple: Find the least common multiple of $4x^2 - 16$ and $6x^2 - 24x + 24$. Solution Step 1 Factor each polynomial. Write ...
2
votes
1answer
44 views

Help with Polynomial Roots Problem

Let's consider the case of two variables, $p\in\mathbb{R}[x,y]$. Suppose I want to find when there is $c\in\mathbb{R}$ such that $$p(x,x)+p(x,c-x)-p(c-x,x)-p(c-x,c-x)=0 \textbf{ for all } ...
1
vote
1answer
82 views

complex numbers, complex roots of equation.

$z_1=a+bi$ , $a,b\in\Bbb R$, $b\neq 0$ is a complex root of the equation $z^2-2z+25=0$. Without evaluating the roots, answer the following questions: i) show that $\overline{z_1}$, the conjugate of ...
4
votes
0answers
190 views

norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
2
votes
3answers
73 views

A question about degree of a polynomial

Let $R$ be a commutative ring with identity $1 \in R$, let $R[x]$ be the ring of polynomials with coefficients in $R$, and let the polynomial $f(x)$ be invertible in $R[x]$. If $R$ is an integral ...
2
votes
1answer
74 views

How to determine the existence of points on a circle (with polynomials)

Let $P\in\mathbb{C}[X]$ and $a\in\mathbb{C}$ such as $P(a)\neq 0$. Assume that $a$ is root of order $k$ of $P-P(a)$. Show, for $\rho>0$ small enough, there exist, on a circle centred at $a$ and ...
0
votes
1answer
88 views

MATLAB: Approximate tomorrow's temperature with 2nd, 3rd and 4th polynomial using the Least Squares method.

The following is Exercise 3 of a Numerical Analysis task I have to do as part of my university course on the subject. Find an approximation of tomorrow's temperature based on the last 23 values ...
3
votes
4answers
91 views

Number of solutions of $P(x)=e^{ax}$ if $P$ is a polynomial

In MSE question the equation $x^2-1=2^x$ is considered, this is a generalization: Let $P_n(x)$ a polynomial of degree $n > 0$. It is well know that the equation $P_n(x)=0\;$ has at most $n$ real ...
0
votes
2answers
146 views

Why doesn't the polynomial factor theorem hold for polynomials in a non-field ring?

I was reading in a book that the Factor Theorem only holds over fields (not rings). Why would that be true? No where in the proof of the factor theorem is a multiplicative inverse taken, so the proof ...
1
vote
1answer
36 views

Proving that $|\Phi_n(x)| > x-1$

Let $\Phi_n$ be the n-th cyclotomic polynomial. I'd like to prove that $$\forall n \geq 2, \forall x \in [2, \infty[, |\Phi_n(x)| > x-1$$ The result is clear when $n$ is prime, but I'm struggling ...
0
votes
4answers
66 views

Recognizing the proper polynomial factorization to solve an indeterminate limit

I had to solve the $\lim_{x \to 3} \frac{x^3-3x^2-x+3}{x^2-x-6}$ that is indeed an indeterminate form ($\frac{0}{0}$). The approach I adopted was to factor the polinomials so that I can deviate from ...
2
votes
1answer
71 views

A field with irreducible polynomial that has multiple roots

Can you give me an example of a field $\mathbb{K}$ such that there exists a polynomial $p(x)\in\mathbb{K}[x]$ that is irreducible and has a multiple root?
3
votes
1answer
826 views

Factoring $(a+b)(a+c)(b+c)=(a+b+c)(ab+bc+ca)-abc$

How to prove the following equality? $$(a+b)(a+c)(b+c)=(a+b+c)(ab+bc+ca)-abc$$ I did it $$\begin{aligned} a^2b + a^2c + ab^2 + cb^2 + bc^2 + ac^2 + 2abc &=a^2(b + c) + bc(b + c) + a(b^2 + ...
3
votes
0answers
56 views

polynomials and functions on $\mathbb{Z}/n\mathbb{Z}$

My general question is How is the set of all polynomial functions on $\mathbb{Z}/n\mathbb{Z}$ structured? What is the number of such functions? How, given a function, one can recognize that it is ...
1
vote
6answers
184 views

How to factorize $2x^2+5x+3$?

I'm doing pre-calculus course at coursera.org and I'm in trouble with this solution $$2x^2 +5x +3 = (2x+3)(x+1)$$ By trial, using ac-method I got stuck: $$ ac = (2)(3) = 6\\ 6 + ? = 5 \Rightarrow~ ? ...
1
vote
1answer
38 views

n-th power over different algebraic structure

It is a classical result that the group $\mathbb{F}_p^{\times}$ is cyclic and that the equation $x^n \equiv a \pmod{p}$ is solvable iff $a^{(p-1)/gcd (p-1,n)} \equiv 1 \pmod{p}$. Also, we know that if ...
0
votes
4answers
120 views

If $(x - 2)$ is a factor of $x^3 + ax^2 -6x -4$, then find $a$.

If $(x - 2)$ is a factor of $x^3 + ax^2 - 6x - 4$, then find $a$. This is regarding polynomials. The answer is $a = 2$. Could someone please provide the working out and help me out on this please. ...
3
votes
1answer
151 views

Proof Verification: The polynomial $f(x) = (x+1)^n-x^n-1$ has a root of multiplicity 2 if and only if $n \equiv 1 \pmod 6$

Proof Verification: The polynomial $f(x) = (x+1)^n-x^n-1$ has a root of multiplicity 2 if and only if $n \equiv 1 \mod 6$ Let $r$ be a root, real or complex, of multiplicity 2 of $f(x)$. Then, by the ...
3
votes
1answer
273 views

Accuracy of the Newton-Cotes formulas for polynomials of degree $n+1$ and even $n$

Let $f$ be a polynomial of degree $n+1$. The Newton-Cotes formula is given by $$\int_{-t}^tf(x)\text{ d}x\approx\sum_{k=0}^nf(x_k)\int_{-t}^t\omega_{n+1}(x)\text{ d}x \tag{*}$$ where ...
2
votes
1answer
32 views

Why holds $\int_{-t}^t\omega_{n+1}(x)\text{ dx}=-\int_{-t}^t\omega_{n+1}(t-x)\text{ dx}$ for the Newton basis polynomials $\omega_{n+1}(x)$

Let $$\omega_{n+1}(x):=\prod_{k=0}^n(x-x_k)$$ denote the Newton basis polynomials and $$x_k:=kh-t\;,\;\;\;h:=2\frac{t}{n}$$ Why holds $$\int_{-t}^t\omega_{n+1}(x)\text{ ...
1
vote
1answer
62 views

How to verify if characteristic equation is right?

I am new to EigenValues and EigenVectors. I am trying to solve a basic sum and somehow I am going wrong. The formula I know to get the characteristic equation is: $\lambda^3 - \sum(\text{diagonal ...
0
votes
2answers
66 views

Find roots of polynomial $f(X) = X^7 - 6 X^6 + 10 X^5 - 13 X^3 + 18 X^2 -22 X + 12 \in \mathbb Q[X]$

Find the roots of the polynomial $$ f(X) = X^7 - 6 X^6 + 10X^5 - 13 X^3 + 18 X^2 -22 X + 12 \in \mathbb Q[X] $$ in $\mathbb Q$, $\mathbb R$ and $\mathbb C$. We covered the factor-theorem in ...
1
vote
1answer
43 views

Find a polynomial whose splitting field is $\mathbb{Q}[\alpha,i]$

Let $f(x)=x^{3}-3x+1$ and let $\alpha$ be a root in $f$. i) Show that the polynomial $f$ is irreducible in $\mathbb{Q}[x]$. ii) Show $\alpha^{2}-2$ is a root of $f$ as well, and show that all roots ...
4
votes
2answers
70 views

Prove two bases are dual in a finite field.

Let K be a finite field, $F=K(\alpha)$ a finite simple extension of degree $n$, and $ f \in K[x]$ the minimal polynomial of $\alpha$ over $K$. Let $\frac{f\left( x \right)}{x-\alpha }={{\beta ...
2
votes
5answers
120 views

Factoring $4x^4 + 12x^3 - 24x^2 - 32x$ [closed]

Some help with factorizing this polynomial please. I have tried but it is difficult as it factorizes down to a cubic and I can't factorize it further. This is regarding the division of polynomials. ...