Question

Polynomials
find all non-Zero polynomials P(x) with real coefficients such that:

\[P(2x)=P'(x) P''(x)\]

Answer 1

I only see one non-constant polly that works

Answer 2

what is that?

Answer 3

\[p(x)=\frac{4}{9}x^3\]

Answer 4

how?

Answer 5

let's suppose that the degree of P(x) is n
hence
degree of P(2x) is : n
degree of P'(x) is : n-1
degree of P''(x) is : n-2
degree of P'(x) P''(x) is : n-1+n-2=2n-3
according to the equation degree of P(2x) is equal to degree of P'(x) P''(x)
so
n=2n-3 --> n=3

then we have
\[P(x)=ax^3+bx^2+cx+d \\ P(2x)=8ax^3+4bx^2+2cx+d \\ P'(x)=3ax^2+2bx+c\\ P''(x)=6ax+2b\]
according to the equation
\[ 8ax^3+4bx^2+2cx+d =(3ax^2+2bx+c)(6ax+2b)\]
simplifying and solve for coefficients gives
a=4/9 and b=c=d=0

then the only answer is
\[P(x)=\frac{4}{9} x^3\]

Answer 6

nice!!!

Answer 7

why do you choose
\[P(x)=ax^3+bx^2+cx+d \]

Answer 8

because we found the degree of polynomial and that was n=3

Answer 9

oh I see now..