Question

Using the Remainder Theorem find the remainders obtained when ` x^3 + (kx + 8 ) x + k ` is divided by x + 1 and x - 2 . 

Hence find k if the sum of the two remainders is 1.

Answer 1

Remainder theorem :
Dividend = Divisors ×  Quotient + Remainder

∴ Let f  ( x ) = `x^3 + (kx + 8 ) x + k`

                 ` = x^3  + kx ^2 + 8x + k`

Dividing f (x ) by x + 1 gives remainder as  R₁

∴  f (-1) = R₁ 

Also , f ( 2 ) = R₂ 

∴  f (-1) = (-1)³ + k (-1)² + 8 (-1) + k

             = -1 + k - 8 + k

             =`2k - 9 = R_1`

`f (2) = (2)^3 + k (2)^2 + 8 xx 2 + k`

       ` = 8 + 4k + 16 + k` 

       ` = 5 k + 24 = R_2`

Also,Sum of remainders = `R_1 +  R_1 = 1`

∴ ( 2k - 9 ) + ( 5k +24 ) =1 

7k + 15 = 1

7k = -14 

k = - 2