Question

find the value of a and b so that x^4+x^ 3+8×^2+ ax=b is divisible by (x^ 2+1)

Answer 1

x^2+1=0

x^2=-1=i^2

x=+/-i

Put x=+i , R=i^4+i^3+8i^2+ai+b=0

1-i-8+ai+b=0

(a-1).i+b=7…………………(1)

Put x=-i

R=(-i)^4+(-i)^3+8(-i)^2-ai+b=0

1-i^3+8i^2-ai+b=0

1+i-8-ai+b=0

(1-a)i+b=7……………………..(2)

Subtract eq.(2)from (1)

2(a-1).i=0

2 and i does not equal zero,

(a-1)=0

a=1 , put a=1 in eq.(1)

b=7

a=1 ,b=7 , Answer