1983 AMC 12 Problem 18

Attempt Problem 18 of the 1983 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1983 AMC 12 solutions, or check the answer key.

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18.

Let

f

be a polynomial function such that, for all real

x,

f(x^2+1)=x^4+5x^2+3.

For all real

x,

f(x^2-1)

x^4+5x^2+1

x^4+x^2-3

x^4-5x^2+1

x^4+x^2+3

none of these

Rewrite the right side as a polynomial in

x^2+1

Setting

t=x^2+1

gives

f(t)=t^2+3t-1

With

t=x^2+1,

\begin{aligned} x^4+5x^2+3 &=(t-1)^2\\ &\quad+5(t-1)+3\\ &=t^2+3t-1. \end{aligned}

Hence

f(t)=t^2+3t-1.

Substituting

t=x^2-1

gives

\begin{aligned} f(x^2-1) &=(x^2-1)^2\\ &\quad+3(x^2-1)-1\\ &=x^4+x^2-3. \end{aligned}

Therefore, the correct answer is B.

Problem 18 in other years