2017 AMC 12A Problems/Problem 23
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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 from Vieta's formulas on the quadratic term of and the cubic term of , we obtain the following:
Now applying Vieta's formulas on the constant term of , the linear term of , and the linear term of , we obtain:
Substituting for and factoring the remainder of the expression, we obtain:
It follows that . But so
Now we can factor in terms of as