# Find all Polynomials P(x) with real coefficients

Find all Polynomials P(x) with real coefficients so that $2P(2x) = P(3x) + P(x)$. I tried to substitute first degree, second and third, bit couldn't get an equality. Thank you for your responses!

• can yo find the constant term? – Jorge Fernández Hidalgo Jan 6 '17 at 16:53
• Look at the leading coefficient. – lhf Jan 6 '17 at 16:56

Let the polynomial be $a_nx^n+\dots+ a_0$

then the leading term of the polynomial on the left is $2^{n+1}a_n$ and the leading term of the polynomial on the right is $(3^n+1)a_n$

So clearly we must have $n=1$ or $0$.

So the polynomial is of the form $ax+b$. and any of these work as we have:

$4ax+2b=3ax+b+ax+b$

• just looking at you equation: either you forgot a factor $a$ for $x$ on the right hand side or $a=1$. – Max Jan 6 '17 at 17:00
• fixed, thanks. @Max – Jorge Fernández Hidalgo Jan 6 '17 at 17:07
• good point, we get $n=0$ or $n=1$. The larger values do not work because the right grows faster. – Jorge Fernández Hidalgo Jan 6 '17 at 17:23

For a term of degree $d$,

$$2a_d(2x)^d=a_d(3x)^d+a_dx^d$$ requires

$$a_d(2^{d+1}-3^d-1)=0.$$

The only non-trivial solutions are with $d=0,d=1$, so that the polynomial is of the first degree.