F.4 Maths

(a) By remainder theorem, since x - 1 is a factor
Therefore f(1) = 0
2 - 1 + a + b = 0
1 + a + b = 0 ... (1)
Remainder is 10 when divided by x + 1
Therefore f(-1) = 10
-2 - 1 - a + b = 10
-13 - a + b = 0 ... (2)
(1) + (2) => -12 + 2b = 0
b = 6
Sub into (1), 1 + a + 6 = 0
a = -7
(b) f(x) = 2x^3 - x^2 - 7x + 6
f(x)/(x - 1) = 2x^2 + x - 6
= (2x - 3)(x + 2)
Therefore f(x) = (x - 1)(x + 2)(2x - 3)