If x² + 1 is a factor of x⁴ +x³ + 8x² +ax +b, find the values of a and b.

Answer :

We have x² +  1 is a factor of our polynomial  x⁴ + x³ + 8x² + ax  + b 

So we take  x² +  1 = 0

x²  = -1

x  =  i  And  - i  ( As we know i  = $\sqrt{-1}$  , where i is a imaginary number )

So our polynomial given zero At x  =  i  And  - i 

First we take x  =  i  , And get

i⁴ + i³ + 8i² + ai  + b  = 0

1 - i  - 8  +  ai  + b  = 0

ai  - i  + b  = 7  ------------------- ( 1 )

Now At x  =  -i  , And get

( - i )⁴ + ( - i )³ + 8( - i )² + a( - i )  + b  = 0

1 + i  - 8  -  ai  + b  = 0

- ai  + i  + b  = 7  ------------------- ( 2 )

Now we  add equation 1 and 2 , we get

2b  =  14

b  =  7  ,  Substitute that value in equation 1 and get

ai  - i  + 7  = 7 

ai  =  i 

a =  1 

So,


we get

a  =  1
And
b  =  7                                                                                             ( Ans  )