2007 AMC 12A Problem 18

Attempt Problem 18 of the 2007 AMC 12A 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 2007 AMC 12A solutions, or check the answer key.

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18.

The polynomial

f(x)=x^4+ax^3+bx^2+cx+d

has real coefficients, and

f(2i)=f(2+i)=0.

What is

a+b+c+d?

0

1

4

9

16

Real coefficients force the conjugates

-2i

and

2-i

to be roots as well

f(x)=(x^2+4)(x^2-4x+5)

; the sum

a+b+c+d

equals

f(1)-1

Since

f

has real coefficients, the conjugates

-2i

and

2-i

are also roots. Thus

\begin{gathered} f(x) = (x^2+4)(x^2-4x+5) \\ = x^4-4x^3+9x^2 \\ {}-16x+20. \end{gathered}

Then

a+b+c+d=-4+9

-16+20

=9.

Equivalently,

a+b+c+d=f(1)-1

=(1+4)(1+1)-1

=9.

Thus, the correct answer is D.

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