Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.2 AIME Find the answer Philippines

Problem:

Three of the roots of $x^{4} + a x^{2} + b x + c = 0$ are $3$ , $-2$ , and $5$ . Find the value of $a + b + c$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

The coefficient of $x^{3}$ is $0$ , so by Vieta's formula, the sum of the roots must be $0$ , and the fourth root must be $-6$ . Therefore, the polynomial factors as $(x-3)(x+2)(x-5)(x+6)$ . Setting $x=1$ gives $1 + a + b + c = (1-3)(1+2)(1-5)(1+6) = 168$ , so $a + b + c = 167$ .

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