2003 AMC 12A Problems/Problem 21
Problem
The graph of the polynomial
has five distinct -intercepts, one of which is at
. Which of the following coefficients cannot be zero?
Solution
Solution 1
According to Vieta's formulas, the sum of the roots of a 5th degree polynomial taken 4 at a time is . Calling the roots
and letting
(our given zero at the origin), the only way to take four of the roots without taking
is
.
Since all of the other products of 4 roots include
, they are all equal to
. And since all of our roots are distinct, none of the terms in
can be zero, meaning the entire expression is not zero. Therefore,
is a sum of zeros and a non-zero number, meaning it cannot be zero, so
.
Solution 2
Clearly, since is an intercept,
must be
. But if
was
,
would divide the polynomial, which means it would have a double root at
, which is impossible, since all five roots are distinct.
See Also
2003 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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