Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.2 AIME Find the answer Philippines

Problem:
The polynomial $p(x) = x^{2} - 3x + 1$ has zeros $r$ and $s$ and a quadratic polynomial $q(x)$ has leading coefficient $1$ and zeros $r^{3}$ and $s^{3}$ . Find $q(1)$ .

Your answer

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

Solution

Solution:
We have $r + s = 3$ and $rs = 1$ . By Vieta's formulas, we compute
$r^{3} + s^{3} = (r + s)^{3} - 3rs(r + s) = 27 - 3(1)(3) = 18$
and $r^{3} s^{3} = 1$ . Thus, we obtain $q(x) = (x - r^{3})(x - s^{3}) = x^{2} - (r^{3} + s^{3})x + r^{3} s^{3} = x^{2} - 18x + 1$ . Hence, we get $q(1) = -16$ .

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