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

(Problem statement and solution to 2017 AMC 12A #23) |
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Now we can factor <math>f(x)</math> in terms of <math>g(x)</math> as <math>f(x)=(x-r_4)g(x)=(x+90)g(x)</math>. Then <math>f(1)=91g(1)</math> and <math>g(1)=1^3-89\cdot 1^2+1+10=-77</math>. | Now we can factor <math>f(x)</math> in terms of <math>g(x)</math> as <math>f(x)=(x-r_4)g(x)=(x+90)g(x)</math>. Then <math>f(1)=91g(1)</math> and <math>g(1)=1^3-89\cdot 1^2+1+10=-77</math>. | ||

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

## Revision as of 15:39, 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 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 .

Hence .