Question

Find the values of m and n when the polynomial f(x)= x³ - 2x² + m x +n has a factor (x+2) and leaves a remainder 9 when divided by (x+1).

Answer 1

(x+2) ⇒  x =- 2 .... (i) 

(x+l) ⇒  x = -1 .... (ii) 

Putting (i) in polynomial, we get 

f(-2) = (-2) × (-2)× (-2) - 2 × (-2) × (-2) + m × (-2) + n = 0 

⇒ -8 -8 - 2m + n= 0 

⇒  n =2 m + 16 .... (iii) 

Putting (ii) in polynomial, and remainder is 9 we get 

f(-1) = (-1) × (-1) × (-1) - 2 × (-1) × (-1) + m × (-1) + n = 9 

⇒ - 1 - 2 - m + n = 9

⇒ m = n - 12    .....(iv)

Combining (iii) and (iv), we get, 

n = 2 x (n - 12) + 16 ,

⇒ n = 8

Hence, m = n - 12 = 8 - 12 = -4 

m = - 4, n = 8