The remainder is not this!
Algebra
Level
3
Let . Find the remainder when is divided by .
3.
5.
2.
0.
1.
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f ( x ) = x 4 + x 3 + x 2 + x + 1
f ( x 5 ) = x 2 0 + x 1 5 + x 1 0 + x 5 + 1
f ( x 5 ) = ( x 2 0 − 1 ) + ( x 1 5 − 1 ) + ( x 1 0 − 1 ) + ( x 5 − 1 ) + 5
Notice that ( x 2 0 − 1 ) is divisible by ( x 5 − 1 ) .
This is also true for ( x 1 5 − 1 ) , ( x 1 0 − 1 ) and ( x 5 − 1 ) .
Also notice that ( x 5 − 1 ) is divisible by f ( x ) .
Hence, the remainder when f ( x 5 ) is divided by f ( x ) is 5 .