Logic - al - jabr

Algebra Level 4

Let $c$ and $d$ be integer constants such that the polynomial $f(x) = x^3 - 8x^2 + cx - d$ has three distinct positive integers as its roots. Find $f(1)$ .

The answer is 0.

1 solution

Otto Bretscher
Feb 29, 2016

Since the sum of the three distinct positive roots of $f(x)$ is 8 (by Viete), one of these roots has to be 1. Note that $2+3+4>8.$ Thus $f(1)=\boxed{0}$ .

Amazing solution!! I was trying to find all possible roots which turned out to be $(1,2,5)$ and $(1,3,4)$ , though in both the cases the answer was $0$ , and it's obvious because in both the cases, the $1$ is a root. But this thought didn't strike at that time. An upvote from me!!!

Raushan Sharma - 5 years, 3 months ago

This is really a very clever solution. I spent time on this problem on computing the value of $c-d$ which is required in computing $f(1)$ . However , I was unable to find it , so I resorted to trial and error.

Nihar Mahajan - 5 years, 3 months ago

Logic - al - jabr

The answer is 0.

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