Question

If x^3+mx^2+nx +6 has x-2 as a factor and leaves a remainder 3, when divided by x-3, find the values of m and n

Answer 1

Let the given polynomial be p(x).

Since, (x - 2) is a factor of p(x)
                                               p(2) = 0
=>  [tex] 2^{3} + m(2)^2 + n(2) + 6 = 0 [/tex]
=>                        8 + 4m + 2n + 6 = 0
=>                             4m + 2n + 14 = 0
=>                                       2m + n = - 7      . . . . . (1)

When p(x) is divided by (x - 3), leaves a remainder 3
                                               p(3) = 3
=>  $3^{3} + m(3)^2 + n(3) + 6 = 6$
=>                      27 + 9m + 3n + 6 = 6
=>                            9m + 3n + 27 = 0
=>                                      3m + n = - 9 . . . . . (2)   
(1) - (2) 
=>            -m = 2
                 m = -2
Then,
                  n = -3

Therefore, m = -2, n = -3