Find the remainder of f divided by g ( x ) = x 4 </msup> + x 2

Mara Cook

Mara Cook

Answered question

2022-06-22

Find the remainder of f divided by g ( x ) = x 4 + x 2 + 1 if the remainder of f divided by h 1 ( x ) = x 2 + x + 1 is x + 1 and the remainder of f divided by h 2 ( x ) = x 2 x + 1 is 3 x + 5

Answer & Explanation

Jovan Wong

Jovan Wong

Beginner2022-06-23Added 23 answers

Let the polynomial be P ( x ). It is given that for some polynomials Q ( x ) , Q 1 ( x )
P ( x ) = ( x 2 + x + 1 ) Q ( x ) + 1 x
P ( x ) = ( x 2 x + 1 ) Q 1 ( x ) + 3 x + 5
Now it is well known that x 2 + x + 1 has the zeros as ω , ω 2 So We get
P ( ω ) = 1 ω
P ( ω 2 ) = 1 ω 2
P ( ω ) = 3 ω + 5
P ( ω 2 ) = 3 ω 2 + 5
Now let us assume for some polynomial Q 2 ( x ) we have
P ( x ) = ( x 4 + x 2 + 1 ) Q 2 ( x ) + A x 3 + B x 2 + C x + D
Using the fact that x 4 + x 2 + 1 has the zeros ω , ω 2 , ω , ω 2 we get four linear equations as:
[ 1 ω 2 ω 1 1 ω ω 2 1 1 ω 2 ω 1 1 ω ω 2 1 ] [ A B C D ] = [ 1 ω 1 ω 2 5 3 ω 5 3 ω 2 ]
Now let us solve by Cramer's rule. The determinant of the matrix is Δ = 12, Δ 1 = 24, Δ 2 = 24, Δ 3 = 12 and Δ 4 = 60. Thus the values of A,B,C,D are
A = Δ 1 Δ = 2 B = Δ 2 Δ = 2 C = Δ 3 Δ = 1 D = Δ 4 Δ = 5
So the required remainder is
2 x 3 + 2 x 2 + x + 5

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