AMC 2007 12A P18

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Problem Statement

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

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Solution

$$f(x) = (x - r_1)(x - r_2)(x - r_3)(x - r_4)$$ $$\text{let } r_1 = 2i \implies r_2 = -2i$$ $$\text{let } r_3 = 2 + i \implies r_4 = 2 - i$$ $$f(x) = (x - 2i)(x - (-2i))(x - (2 +i))(x - (2 - i))$$ $$f(x) = (x^2 - 4i^2)(x^2 - 4x + 4 - i^2) = (x^2 + 4)(x^2 - 4x + 5)$$ $$f(x) = x^4 - 4x^3 + 5x^2 + 4x^2 - 16x + 20$$ $$f(x) = x^4 - 4x^3 + 9x^2 - 16x + 20$$ $$\therefore a = -4, b = 9, c = -16, d = 20 \implies a+b+c+d = 9$$