What is the value of i ( i − 1 ) ( i − 2 ) ( i − 3 ) i ( i − 1 ) ( i − 2 ) ( i − 3 ) i = − 1 ? i = − 1 ?
Answer Choices:
A. 6 − 5 i 6 − 5 i − 10 i − 1 0 i 10 i 1 0 i − 10 − 1 0 10 1 0
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Consider the polynomial:
P ( x ) = x ( x − 1 ) ( x − 2 ) ( x − 3 ) = x 4 − 6 x 3 + 11 x 2 − 6 x P ( x ) = x ( x − 1 ) ( x − 2 ) ( x − 3 ) = x 4 − 6 x 3 + 1 1 x 2 − 6 x
Then P ( i ) = i ( i − 1 ) ( i − 2 ) ( i − 3 ) = 1 + 6 i − 11 − 6 i = − 10 P ( i ) = i ( i − 1 ) ( i − 2 ) ( i − 3 ) = 1 + 6 i − 1 1 − 6 i = − 1 0 (D) -10 (D) -10
Alternate Solution:
We can manually multiply everything out as follows:
i ( i − 1 ) ( i − 2 ) ( i − 3 ) = ( − 1 − i ) ( i − 2 ) ( i − 3 ) = ( 3 + i ) ( i − 3 ) = − 10 i ( i − 1 ) ( i − 2 ) ( i − 3 ) = ( − 1 − i ) ( i − 2 ) ( i − 3 ) = ( 3 + i ) ( i − 3 ) = − 1 0
which yields the same answer as before.
The problems on this page are the property of the MAA's American Mathematics Competitions