Find the remainder
Let f ( x ) = x 4 + x 3 + x 2 + x + 1 . Find the remainder when f ( x 5 ) is divided by f ( x ) .
Source : Problem solving strategies by Arthur Engel.
The answer is 5.
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3 solutions
for x=1 , remainder is 0
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for x= 0 , the remainder is 0
However, we are considering the polynomial remainder, which works generally.
What is the difference between polynomial remainder and the normal remainder?
manual FTW!!!!!!!!!!
Yeah, very neat solution !
It is given that f ( x ) = x 4 + x 3 + x 2 + x + 1
Therefore, f ( x 5 ) = x 2 0 + x 1 5 + x 1 0 + x 5 + 1
We have ( 1 − x ) ( x 4 + x 3 + x 2 + x + 1 ) = 1 − x 5
⇒ x 5 = 1 + ( x − 1 ) ( x 4 + x 3 + x 2 + x + 1 ) = 1 + ( x − 1 ) f ( x )
It should be noted that x 5 n ≡ 1 m o d f ( x )
Therefore, the remainder of f ( x ) x 2 0 + x 1 5 + x 1 0 + x 5 + 1 is 5 .
Use Factor theorem by observing that the fifth root of unity is a root of f ( x )
Firstly, ( x − 1 ) f ( x ) = x 5 − 1
Also, x 5 n − 1 = ( x 5 − 1 ) ( x 5 n − 5 + x 5 n − 1 0 + . . . + 1 ) for all positive integers n hence x 5 − 1 divides x 5 n − 1 .
Now this means f ( x ) divides x 5 − 1 which divides ( x 2 0 − 1 ) + ( x 1 5 − 1 ) + ( x 1 0 − 1 ) + ( x 5 − 1 ) + ( 1 − 1 ) = f ( x 5 ) − 5 . Hence the remainder is 5.