2019 AMC 10A Problems/Problem 24
Problem
Let ,
, and
be the distinct roots of the polynomial
. It is given that there exist real numbers
,
, and
such that
for all
. What is
?
Solution 1
Multiplying both sides by yields
As this is a polynomial identity, and it is true for infinitely many
, it must be true for all
(since a polynomial with infinitely many roots must in fact be the constant polynomial
). This means we can plug in
to find that
. Similarly, we can find
and
. Summing them up, we get that
By Vieta's Formulas, we know that
and
. Thus the answer is
.
Note: This process of substituting in the 'forbidden' values in the original identity is a standard technique for partial fraction decomposition, as taught in calculus classes.
Solution 2 (limits)
Multiplying by on both sides, we find that
As
, notice that the
and
terms on the right will cancel out and we will be left with only
. Hence,
, which by L'Hôpital's rule becomes
. We can reason similarly to find
and
. Adding up the reciprocals and using Vieta's Formulas, we have that
See Also
2019 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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