Step 1: Form the equations
Given: f(x)=x3−2x2+mn+mn
A polynomial f(x) has a factor (x-a) if and only if f(a)=0.
f(x) has a factor (x + 2).
∴f(-2)=0
(−2)3−2(−2)2+m(−2)+n=0
−8−8−2m+n=0
n=2m+16-(i)
Also, f(x) leaves remainder 9 when divided by (x + 1)
∴By remainder theorem, when a polynomial f(x) is divided by f(x-a), then the remainder is f(a).
∴f(-1)=9
(−1)3−2(−1)2+m(−1)+n=9
−1−2−m+n=9
n−m=12-(ii)
Step 2: Find the value of m and n
Put the value of n in equation (ii)
2m+16−m=12
m=−4
Put the value of m in equation(ii)
n=2(−4)+16
n=8
Hence, m=−4 and n=8 .