Let's go to the polynomials

Calculus Level 4

$P(x)$ is a non-zero polynomial with real coefficients and satisfies following equation $P(2x)=P'(x)P''(x)$ Where $P'(x)$ and $P''(x)$ are first and second derivatives of $P(x)$ respectively. Evaluate $P(3)$ .

The answer is 12.

2 solutions

Kazem Sepehrinia
May 26, 2015

Let degree of $P(x)$ be $n$ , note that $\deg(P(2x))=\deg(P'(x) P''(x)) \\ n=n-1+n-2 \\ n=3$ So $P(x)$ is a cubic polynomial and for finding coefficients of polynomial one can write $a(2x)^3+b(2x)^2+c(2x)+d=(3ax^2+2bx+c)(6ax+2b)$ Solving this gives us $a=\frac{4}{9}, \ b=c=d=0$ Therefore, $P(x)=\frac{4}{9} x^3$ and $P(3)=12$

Raghav Vaidyanathan
Apr 14, 2015

Due to obvious reasons, the answer is:

$P(x)=\frac {4} {9} x^3$

(obtained by intuitively substituting $P(x)=kx^3$ and then solving for value of $k$ ).

Moderator note:

You are right that $P(x) = \frac 4 9 x^3$ only. But you didn't prove that it is unique.

Let's go to the polynomials

The answer is 12.

2 solutions

Moderator note:

0 pending reports