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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10
votes
1answer
90 views

For a fixed degree, is there always a Lagrange polynomial below the original function?

Let $x_1<x_2< \ldots <x_n$ be $n$ real numbers, and let $y_1,y_2,\ldots,y_n$ be real values to be interpolated. Let $r\leq n$. For any $I\subseteq \lbrace 1,2,\ldots,n\rbrace$ of cardinality ...
2
votes
2answers
964 views

Is there a rule to the terms of a falling factorial?

$\require{cancel}$I discovered that $n!=\xcancel{(n)_{n-1}}n^{\underline{n-1}}=n(n-1)(n-2)\cdots(3)(2)$. I have expanded a few examples: $$2!=\xcancel{(2)_1}2^{\underline{1}}=2\\ ...
1
vote
0answers
45 views

Ordered field of Polynomials.

List the following functions from smallest to largest. $$\frac{x^2 + 2}{x-1},\frac{x^2 - 2}{x+1},\frac{x + 1}{x^2-2},\frac{x + 2}{x^2-1}$$ To solve the problem, I believe we compare each fraction ...
6
votes
2answers
231 views

Why Rational Root Theorem only works with integers

Why does the rational root theorem only work when the polynomial has integer coefficients?
0
votes
2answers
88 views

Factorization of a polynomial

I need to find the roots of this polynomial $$2x^2-x^4-x=0.$$ I know that it is necessary the factorization to obtain $$-x(x-1)(x^2+x-1)=0.$$ I asked to factorize my polynomial to Mathematica. The ...
8
votes
4answers
758 views

Understanding non-solvable algebraic numbers

Background We know from Galois theory that the zeros of a polynomial with rational coefficients whose Galois group is solvable can be expressed in a formula that involves rational powers of the ...
0
votes
2answers
80 views

Finding the maximum and minimum of $V(t) = 2t^2 − 16 t + 40$

The volume of water in a tank, V m3, over a 10 month period is given by the function $V(t) = 2t^2 − 16 t + 40,$ where t is in months and $ t ∈ [0, 10].$ I completed the sure and got $2(t - 4)^2 + 24$ ...
0
votes
3answers
41 views

$f(x) = 5 + 6x − 3x^2, \ x \in [−5, 3)$

$$f(x) = 5 + 6x − 3x^2,\ x \in [−5, 3)$$ Sketch the graph of each of the functions below and state the domain and the range of each function. I found the $y$ intercept which is $5$. But the $x$ ...
2
votes
1answer
165 views

Using Polya's Theorem to check positivity of a multivariate polynomial

I wish to check if a homogeneous polynomial of total degree 4 is positive definite. The polynomial is of the form $$P(u,v,x,y) = \sum_i\alpha_iu^{i_1}v^{i_2}x^{i_3}y^{i_4}$$ with $0 \le i_j \le 2$, ...
0
votes
3answers
65 views

Show that a set of polynomials make a linear space.

I have a problem that states: "Let P be the set of all polynomials of degree at most 2. Show that P is a linear space." I know how to show that a set of vectors make a linear space with a certain ...
0
votes
1answer
68 views

General term of a series

I am trying to find the general term of the series: $$( 1 + x + ... + x^{m-1} )^k$$ I am trying to implement the KZ filter and it requires the coefficients of the above series. Here, k and m are ...
5
votes
4answers
705 views

Question about Axler's proof that every linear operator has an eigenvalue

I am puzzled by Sheldon Axler's proof that every linear operator on a finite dimensional complex vector space has an eigenvalue (theorem 5.10 in "Linear Algebra Done Right"). In particular, it's his ...
8
votes
2answers
398 views

How to find all polynomials P(x) such that $P(x^2-2)=P(x)^2 -2$?

I am trying the fallowing exercise : Solve $P(X^2 -2)=P(X)^2 -2$ with P a monic polynomial (non-constant) My attempt : Let P satisfying $P(X^2-2) = (P(X))^2-2$ Then $Q(X)=P(X^2-2) = (P(X))^2-2$ ...
-1
votes
1answer
33 views

Find the values in this polynomial

Could I please have help with this polynomial. If (2x-3) and (x+2) are factors of 2x$^3$ + a$x^2$ +bx + 30, find the values of a and b. So x = 3/2 or x = -2 When x = -3/2 = 9/4a + 3/2b + 147/4 ...
0
votes
4answers
383 views

Find the value of $a$ and $b$ if $x+1$ and $x−2$ are factors of $ax^3−4x^2+bx−12$

Find the value of $a$ and $b$ if $x+1$ and $x−2$ are factors of $ax^3−4x^2+bx−12$ Attemption $$ \begin{cases} f(-1) = -a - b - 16\\ f(2) = 8a + 2b - 28 \end{cases} $$ But then when I plug ...
0
votes
0answers
48 views

Find the value of a and b if $( x + 1 )$ and $( x -2 )$ are factors of $ ax^3 - 4x^2 + bx - 12$ [duplicate]

Find the value of $a$ and $b$ if $( x + 1 )$ and $( x -2 )$ are factors of $ax^3 - 4x^2 + bx - 12$. This is regarding polynomials. Thank you
1
vote
1answer
78 views

Division algorithm for polynomials condition on field

I see the following theorem Let $F$ be a field. Suppose $a(x),b(x)\in F[x]$ with $b(x)\neq 0$, then there exists $q(x),r(x)\in F$ such that $a(x)=b(x)q(x)+r(x)$, and either $r(x)=0$ or $\deg ...
1
vote
1answer
80 views

Can this quartic equation be reformulated as $x =$ an expression?

$c = \sqrt{a^2 + x^2} + \sqrt{(b - x)^2 + (a - x)^2}$ Reformulated as a quartic equation: $x^4 + (-a - b)x^3 + (a^2 + ab + 2b^2 - c^2)x^2 + (-ab^2 - b^3)x + (-a^2c^2 + b^4 + c^4) = 0$ Is there an ...
3
votes
1answer
89 views

Monic polynomial reducible in rationals

Let $P(x)\in \mathbb{Z}[x], Q(x),R(x)\in \mathbb{Q}[x]$, and all three polynomials are monic. Suppose $P(x)=Q(x)R(x)$. Is it true that $Q(x),R(x)\in\mathbb{Z}[x]$? Gauss's Lemma says that since ...
1
vote
1answer
96 views

About the notation $\mathbb{Z}[x]/(f(x),p)$

Let $f(x)\in \mathbb{Z}[x]$ be a polynomial and $p$ be a prime. What does the notation $\mathbb{Z}[x]/(f(x),p)$ mean? Is it $\mathbb{Z}/p\mathbb{Z}[x]/(f(x))$ ?
0
votes
1answer
42 views

Proof that complex conjugate of polynomial result equals pynomial result with complex conjugated argument

This question feels uneasy to be expressed by words for me, however, I'm asked to prove this: $$P(\overline{a+bi}) = \overline{P(a+bi)}$$ Of course, $\overline{a+bi} = a-bi$.
3
votes
1answer
152 views

Help with Spivak Calculus Ch3 Problem 6a

Yet again I find myself stuck on a Spivak question. This time it is simply the question that isn't clear to me. It states: If $x_1, ..., x_n$ are distinct numbers, find a polynomial function $f_i$, ...
2
votes
2answers
441 views

Does Fermat's Little Theorem work on polynomials?

Let $p$ be a prime number. Then if $ f(x) = (1+x)^p$ and $g(x) = (1+x)$, then is $f \equiv g \mod p$? I'm trying to prove that for integers $a > b > 0$ and a prime integer $p$, ${pa\choose b} ...
1
vote
2answers
29 views

All monic polynomials of degree $d$ such that $f(x) | f(x^n) \forall n \in \mathbb{Z}^+$?

The coefficients may be complex. I was doing a problem for $d=4$ and am wondering if this can this problem be generalized for any $d$
9
votes
4answers
373 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 ...
3
votes
1answer
88 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
69 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
770 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 ...
9
votes
5answers
317 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
105 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
vote
2answers
210 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
113 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
58 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
votes
2answers
183 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
66 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
votes
0answers
183 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
305 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
76 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
262 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
236 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
104 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
63 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
307 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
119 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
100 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 ...
4
votes
3answers
5k 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
158 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
103 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
85 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
vote
2answers
275 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 ...