Find all polynomials with real coefficients, for which the equality
P(2P(x))=2P(P(x))+2(P(x))2
holds for any real number x.
If the highest order term of P(x) is ax^3, then
the Left hand side's highest order term is a(2ax^3)^3 = 8(a^4)x^27, and
the Right hand side's highest order term is 2a(ax^3)^3 = 2(a^4)x^27.
So a must equal 0.
If the highest order term of P(x) is bx^2, then
the Left hand side's highest order term is b(2bx^2)^2 = 4(b^3)x^4, and
the Right hand side's highest order term is 2b(bx^2)^2 + 2(bx^2)^2) = 4(b^3)x^4 + 2(b^2)x^4
So 4b^3 must equal 4b^3 + 2b^2. So b must equal 0.
If the highest order term of P(x) is cx, then
the Left hand side's highest order term is c(2cx) = 2c^2x, and
the Right hand side's highest order term is 2c(cx)+2(cx)2 = 2c^2(x+x^2)
because these are of a different order, c must be 0
So P(x) = d
2d = 2d+ 2d^2
d = 0
Unless I have made a mistake (very possible), the only polynomial that works is P(x) = 0
Algebraic attempt
