1987 AMC 12 第 24 题

Compare the degrees and leading coefficients of the three polynomials

After proving $f$ is a monic quadratic, compare coefficients in $f(x^2)=[f(x)]^2$

Let $f$ have degree $n\ge1$ and leading coefficient $a.$ The three expressions have degrees $2n,2n,n^2,$ so $n^2=2n$ and therefore $n=2.$ Comparing leading coefficients in $f(x^2)=[f(x)]^2$ gives $a=a^2,$ hence $a=1.$

Write $f(x)=x^2+bx+c.$ Then $$x^4+bx^2+c=(x^2+bx+c)^2.$$ The cubic coefficient forces $b=0,$ and then the quadratic coefficient forces $c=0.$ Thus the only candidate is $f(x)=x^2,$ which indeed satisfies all three expressions. There is exactly one function.

Thus the correct answer is B.