let f(x) be a polynomial of degree n
f(x) = axⁿ + g(x)
where g(x) is a polynomial of degree n-1
we are told
(f(x))² = f(x²)
so
[axⁿ+g(x)]²=a²x²ⁿ + 2axⁿ . g(x) +(g(x))² =ax²ⁿ+ g(x²)

looking at the x² term, we can see that a must be 1
by our definition g(x) is a lower order polynomial that xⁿ
so there is no g(x) for which
2axⁿ . g(x) +(g(x))² = g(x²)
hence g(x) must necessarily be zero and our polynomials will be:

xⁿ, x^^n-1...x⁰

that is n+1 of them, but if you consider x⁰ a zero polynomial, then there are n of them

View other answers by: bugzpodder|slayerchange|PiDeltaPhi(author)