0
votes
1answer
41 views

Writing roots of f(x) as f(a) for some a

I was solving a problem when this random thought popped into my head. Suppose you have a function, say $f(x)=x^2-1=(x-1)(x+1)$. The roots of this function are $-1$ and $-1$. We can write these roots ...
0
votes
0answers
12 views

unknown function: calculation of coefficients in series expansion up to a given degree

I am trying to solve an functional equation of unknown $h\mapsto h(x)$ ($x\in\mathbb{R}$, in the neighbourhood of $0$): $$\mathcal{F}(h)=\mathcal{G}(h) \qquad (*)$$ (assume $\mathcal{F}$ and ...
0
votes
2answers
30 views

Equation Theory: A polynomial with specific remainders when divided by specific divisors. What is the remainder when divided by BOTH divisors

Again, for my Equation Theory class, I have the subject question.$p(x)$ has a remainder of 3 when divided by $x-1$ and a remainder of 5 when divided by $x-3$. What is the remainder when $p(x)$ is ...
4
votes
4answers
92 views

Polynomials that satisfy $(x-1)(p(x+1))=(x+2)(p(x))$ where $p(2)=12$?

I am taking a graduate class on Equation Theory and one of my homework questions asks me to "Determine all polynomials $p(x)$ such that $(x-1)(p(x+1))=(x+2)(p(x))$ and $p(2)=12$. A provided hint is to ...
2
votes
0answers
23 views

Is determining a non-constant solution to a functional inequality with polynomial arguements decidable?

So suppose we have a functional inequality with polynomial arguments in $2$ variables, $\sum_i c_i f(p_i(x,y)) \geq 0$, where $c_i$ are say integer constants and $p_i$ are polynomials, say with ...
1
vote
3answers
43 views

Find all polynomials p with real coefficients

Find all polynomials $p$ with real coefficients such that $p(x+1)=p(x)+2x+1$. I feel like in this question you let $x+1=x'$.
4
votes
1answer
97 views

determine all polynomials $P(x)$ such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial. clearly we have to show $(x+1)P(x-1)-(x-1)P(x)=c$ for all values of $x$ ($c$ is a ...
2
votes
1answer
81 views

Find $g(x)$ if $f(g(x))=f(x)g(x)$ and $g(2)$=37, $f(x)$ and $g(x)$ are polynomials

Suppose $f(x)$ and $g(x)$ are non-zero polynomials with real coefficients, such that $f(g(x))=f(x)\times g(x)$. If $g(2)=37$, find $g(x)$. I tried plugging $f(x)$ and $g(x)$ as $n$ and $m$ ...
8
votes
2answers
271 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 $Q(X)=P(X^2-2) = (P(X))^2-2$ ...
0
votes
1answer
27 views

Finding polynomials sattisfying $P\bigr(-c + K/(u+c)\bigl) (u+c)^2/K =P(u)$

Is there any simple way to find the polynomials satisfying the functional relation \begin{align*} P\left(-c + \frac{K}{u+c}\right) \frac{(u+c)^2}{K} = P(u) \tag{*} \end{align*} Where $K = ...
3
votes
4answers
451 views

Let $f(x)$ be a 3rd degree polynomial such that $f(x^2)=0$ has exactly $4$ distinct real roots

Problem : Let $f(x)$ be a 3rd degree polynomial such that $f(x^2)=0$ has exactly four distinct real roots, then which of the following options are correct : (a) $f(x) =0$ has all three real roots ...
4
votes
2answers
233 views

Polynomials $P(x)$ satisfying $P(2x-x^2) = (P(x))^2$

I am looking (to answer a question here) for all polynomials $P(x)$ satisfying the functional equation given in the title. It is not hard to notice (given that one instinctively wants to complete the ...
1
vote
2answers
92 views

Determine the polynomials $p(x)$ satisfying $x\cdot p(x-1) = (x-26)\cdot p(x)$

Determine the polynomials $p(x)$ satisfying $x\cdot p(x-1) = (x-26)\cdot p(x)$. My Solution: Put $x=0$, we get $p(0) = 0$, Similarly put $x=26,$ we get $p(26) = 0$. That means $x=0,26$ are two roots ...
7
votes
2answers
215 views

Find all real polynomials $p(x)$ that satisfy $\sin( p(x) ) = p( \sin(x) )$

Find all real polynomials $p(x)$ that satisfy $\sin( p(x) ) = p( \sin(x) )$ and prove they are indeed all. Is there an easy way to prove this?
3
votes
1answer
159 views

Find all polynomials $P(x)$ satisfying this functional equation

Find all polynomials $P(x)$ which have the property $$P[F(x)]=F[P(x)], \quad P(0) = 0$$ where $F(x)$ is a given function with the property $F(x)>x$ for all $x\geq 0$. This is an exercise ...
7
votes
1answer
287 views

How to find all polynomials $P(x)$ such that $P(x^2+2x+3)=[P(x+3)]^2$?

Can someone please show me how to: Find all polynomials $P(x)$ such that $P(x^2+2x+3)=[P(x+3)]^2$? I've tried substitiuting $x=0,1$. Can't seem to figure it out. The square on the RHS is confusing ...
14
votes
4answers
540 views

Polynomial: $p(x) = p(x+3)$.

Determine polynomial $p(x)$ s.t. $p(x) = p(x+3)$. Just by looking at the above equation, it immediately appears that p has got to be some kind of constant function. I thought it might also be a ...
23
votes
6answers
2k views

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
2
votes
2answers
312 views

Correct order of books for a beginner

what should be the order of the books in which a beginner should do the following books in algebra: -1.E.J. Barbeau POLYNOMIALS -2. Polynomials and Polynomial Inequalities (Graduate Texts in ...
3
votes
2answers
347 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
290 views

Iterated polynomial problem

Polynomial $P$ satisfies $P(n)>n$ for all positive integers $n$. Every positive integer $m$ is a factor of some number of the form $P(1),P(P(1)),P(P(P(1))),\ldots $. Prove that $P(x)=x+1$.