In connection with
Evaluate
1 5 x 3 + 4 5 x 2 + 1 0 x − 1 0 0 ( m o d x + 5 )
The answer is -900.
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2 solutions
Great ⌣ ¨
1 5 x 3 + 4 5 x 2 + 1 0 x − 1 0 0 = 1 5 x 3 + 7 5 x 2 − 3 0 x 2 − 1 5 0 x + 1 6 0 x + 8 0 0 − 9 0 0 = 1 5 x 2 ( x + 5 ) − 3 0 x ( x + 5 ) + 1 6 0 ( x + 5 ) − 9 0 0 = ( 1 5 x 2 − 3 0 x + 1 6 0 ) ( x + 5 ) − 9 0 0
Therefore,
( 1 5 x 3 + 4 5 x 2 + 1 0 x − 1 0 0 ) % ( x + 5 ) = − 9 0 0
Great ⌣ ¨
By Remainder Theorem , we know that that remainder is essentially equal to f ( − 5 ) , where f ( x ) = 1 5 x 3 + 4 5 x 2 + 1 0 x − 1 0 0 .
It turns out that f ( − 5 ) = − 9 0 0 and hence the answer.