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 $a+b+c+d?$

$0$

$1$

$4$

$9$

$16$

Answer: D

Concepts:complex number polynomial

Difficulty rating: 1630

Solution:

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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