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 16:58, 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 .