Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.2 AIME, harder Prove it Philippines

Problem:

$x^{2} + 4x + 8 = 0$ has roots $a$ and $b$ . Find a quadratic polynomial with integer coefficients whose roots are $\frac{1}{a}$ and $\frac{1}{b}$ .

Solution

Solution:

(ans. $8x^{2} + 4x + 1$ .
$a b = 8$ , $a + b = -4 \Rightarrow \frac{1}{a} \frac{1}{b} = \frac{1}{8}$ , $\frac{a + b}{a b} = \frac{1}{a} + \frac{1}{b} = -\frac{1}{2}$ . This means that the reciprocals are roots of the polynomial $x^{2} + \frac{1}{2}x + \frac{1}{8}$ and hence of $8x^{2} + 4x + 1$ .)

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