2007 AMC 12A Problems/Problem 18

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Solution

The polynomial $f(x) = x^{4} + ax^{3} + bx^{2} + cx + d$ has real coefficients, and $f(2i) = f(2 + i) = 0.$ What is $a + b + c + d?$

$\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ 4 \qquad \mathrm{(D)}\ 9 \qquad \mathrm{(E)}\ 16$

Solution

A fourth degree polynomial has four roots. Since the coefficients are real, the remaining two roots must be the complex conjugates of the two given roots, namely $2-i,-2i$ . Now we work backwards for the polynomial:

$(x-(2+i))(x-(2-i))(x-2i)(x+2i) = 0$

$(x^2 - 4x + 5)(x^2 + 4) = 0$

$x^4 - 4x^3 + 9x^2 - 16x + 20 = 0$

Thus our answer is $- 4 + 9 - 16 + 20 = 9\ \mathrm{(D)}$ .

2007 AMC 12A (Problems • Resources)
Preceded by Problem 17	Followed by Problem 19
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2007 AMC 12A Problems/Problem 18

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