Find the remainder when x <mrow class="MJX-TeXAtom-ORD"> 1001 </mrow> </msup

Laila Andrews

Laila Andrews

Answered question

2022-05-16

Find the remainder when x 1001 is divided by x 4 + x 3 + 2 x 2 + x + 1.

Answer & Explanation

Bentyrchjurvk

Bentyrchjurvk

Beginner2022-05-17Added 15 answers

For a start
q ( x ) := x 4 + x 3 + 2 x 2 + x + 1 = ( x 2 + 1 ) 2 + x ( x 2 + 1 ) = ( x 2 + 1 ) ( x 2 + x + 1 )
Now write x 1001 = k ( x ) q ( x ) + a x 3 + b x 2 + c x + d ( )
where a , b , c , d are real. Now note that q ( i ) = q ( ω ) = 0 where ω = 1 + i 3 2 is a 3rd root of unity.
Since ω 2 = ω 1 we have
ω = c b ω + c ω
so c=0 and b=1 and thus remainder is
x 3 + x 2 + 1

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