2

Find the polynomial $P(x)$ satisfy $P(1)=5$ and $$P(x)=\sqrt[3]{4P(x^3+1)-20}+3, \forall x\in \mathbb{R}$$ I found out that the coefficient of the highest degree is 2, I tried finding the degree but it returns always true.

One result I found out is $P(x)=2x+3$, but I don't know whether this is the only polynomial that satisfies or not.

Trường Hưng Nguyễn
  • 151
  • 2
    Did you find $2x+3$ by trial and error? Also note that any such polynomial must probably be unique : since $P(1) = 5$, this fixes $P(1^3+1) = P(4)$ since $(P(1)-3)^3 = 4P(4)-20$. Then $P(4^3+1) = P(65)$ is determined, and so on. Basically, any polynomial satisfying these conditions is determined on an infinite set, and is hence unique. – Sarvesh Ravichandran Iyer Sep 28 '21 at 12:17

1 Answers1

0

$P(0)=3$ and $P(2)=7$ as seen by putting $x=0$ and $x=1$. If you already know that $P$ has degree at most $2$ then $P$ and $Q(x)=2x+3$ have degree at most $2$ and they agree at the points $0,1,2$. This implies that $P(x)=2x+3$ for all $x$.

Kavi Rama Murthy
  • 311,013