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

m (→Solution: fixing typos and improving structure) |
m (→Solution: chose correct answer choice letter) |
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<cmath>g(1)=1^3-89\cdot 1^2+1+10=-77</cmath> | <cmath>g(1)=1^3-89\cdot 1^2+1+10=-77</cmath> | ||

− | Hence <math>f(1)=91\cdot(-77)=\boxed{\textbf{( | + | Hence <math>f(1)=91\cdot(-77)=\boxed{\textbf{(C)}\,-7007}</math>. |

## Revision as of 15:58, 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:

so .

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

Then and

Hence .