# Difference between revisions of "2017 AMC 12A Problems/Problem 23"

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\end{align*}</cmath> | \end{align*}</cmath> | ||

− | Substituting for <math>r_1r_2r_3</math> and factoring the remainder of the expression, we obtain: | + | Substituting for <math>r_1r_2r_3</math> in the bottom equation and factoring the remainder of the expression, we obtain: |

<cmath>-10+(r_1r_2+r_2r_3+r_3r_1)r_4=-10+r_4=-100</cmath> | <cmath>-10+(r_1r_2+r_2r_3+r_3r_1)r_4=-10+r_4=-100</cmath> |

## Revision as of 17:04, 8 February 2017

## Problem

For certain real numbers , , and , the polynomial has three distinct roots, and each root of is also a root of the polynomial What is ?

## Solution

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:

Thus .

Now applying Vieta's formulas on the constant term of , the linear term of , and the linear term of , we obtain:

Substituting for in the bottom equation and factoring the remainder of the expression, we obtain:

It follows that . But so

Now we can factor in terms of as

Then and

Hence .