2017 AMC 12A Problems/Problem 23
For certain real numbers , , and , the polynomial has three distinct roots, and each root of is also a root of the polynomial What is ?
Let and be the roots of . Let be the additional root of . Then by Vieta's formulas, and so .
Also, Vieta's formulas tell us that and .
Hence so that . But so .
Now we can factor in terms of as . Then and .