Question

If (x2

– 4) is a factor of (mx3

– x2

– 2x + n), then find the value of (2m + n).

Answer 1

(x² - 4) is a factor of (mx³ - x² - 2x + n) .

first of all all factorize (x² - 4) or, (x² - 2²)
we know, a² - b² = (a - b)(a + b) use it here,
(x² - 4) = (x -2)(x + 2)

e.g.,(x - 2)(x + 2) is factor of (mx³ -x² - 2x + n)
or, we can say that x = -2 and +2 are zeros of (mx³ - x² - 2x + n) .

now, put x = -2
m(-2)³ - (-2)² - 2(-2) + n = 0
-8m - 4 + 4 + n = 0
-8m + n = 0 ---------(1)

now, put x = +2
m(2)³ - (2)² - 2(2) + n = 0
8m - 4 - 4 + n = 0
8m + n = 8 --------(2)

solve equations (1) and (2)
equation (1) + equation (2)
2n = 8 => n = 4 put it in equation (1)
m = 1/2

hence, m = 1/2 and n = 4 so, (2m + n) = 2 × 1/2 + 4 = 1 + 4 = 5

Answer 2

this is the correct answer