F.4 Maths

6x^3 - 19x^2 + ax + b = P(x) * (3x^2 - 11x - 4) + 3x - 2

6x^3 - 19x^2 + (a - 3)x + b + 2 = P(x) * (x - 4)(3x + 1)

6x^3 - 19x^2 + (a - 3)x + b + 2 = P(x) * 3(x - 4)(x + 1/3)

By remainder theorem :

6(4^3) - 19(4)^2 + (a - 3)(4) + b + 2 = 0

384 - 304 + 4a - 12 + b + 2 = 0

4a + b + 70 = 0...(1)

6(-1/3)^3 - 19(-1/3)^2 + (a - 3)(-1/3) + b + 2 = 0

-2/9 - 19/9 + (3 - a)/3 + b + 2 = 0

-2 - 19 + 9 - 3a + 9b + 18 = 0

3a - 9b - 6 = 0

a - 3b - 2 = 0.....(2)

(2) + (1)*3 :

13a + 210 - 2 = 0

a = - 16

sub to (2) : -16 - 3b - 2 = 0

b = - 6