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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).
Solution
(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
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