Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME, harder Find the answer

A polynomial $P$ has four roots, $\frac{1}{4}, \frac{1}{2}, 2,4$ . The product of the roots is 1, and $P(1)=1$ . Find $P(0)$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

A polynomial $Q$ with $n$ roots, $x_{1}, \ldots, x_{n}$ , and $Q\left(x_{0}\right)=1$ is given by $Q(x)=\frac{\left(x-x_{1}\right)\left(x-x_{2}\right) \cdots\left(x-x_{n}\right)}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right) \cdots\left(x_{0}-x_{4}\right)}$ , so $P(0)=\frac{1}{\frac{3}{4} \cdot \frac{1}{2} \cdot(-1) \cdot(-3)}=\frac{8}{9}$ .

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