Question

If x^2-4 is a factor of mx^3-x^2-2x+n find the values of m and n

Answer 1

x^2 - 4 is a factor then x^2 - 4 = 0
(x - 2) (x + 2) = 0
x = 2 and x = - 2
Put the value of x one by one in equation
FOR X = 2
m(2)^3 - (2)^2 - 2(2) + n = 0
8m - 4 - 4 +n = 0
8m + n = 8 ......... (1)

FOR X = - 2
m(-2)^3 - (-2)^2 - 2(-2) + n = 0
-8m - 4 + 4 + n = 0
8m = n .......... (2)

Subsitute (2) in (1)
8m + 8m = 8
16m = 8
m = 8 / 16
m = 1 / 2

Place the value in (2)
8 (1/2) = n
n = 4

Therefore, m = 1/2 & n = 4

Answer 2

if x²-4 is a factor of polynomial mx³-x²-2x+n

then
polynomial will be zero at x=±2

so
=> 8m-4-4+n = 0
8m+n = 8

=> -8m-4+4+n =0
n = 8m

so
2n = 8
n = 4

and
m = 1/2


m = 1/2, n = 4