F4 Maths Remainder Theorem 3

1. Find the remainder when P(x) = 32x^5 - 1 is divided by 4x^2 - 1.
Let P(x) = Q(x)(2x + 1)(2x - 1) + ax + b
P(1/2) = 1 - 1 = 0 = a/2 + b ... (1)
P(-1/2) = -1 - 1 = -2 = -a/2 + b ... (2)
(1) + (2) => 2b = -2
b = -1
Sub into (1), 0 = a/2 - 1
a = 2
The remainder is 2x - 1
2. Find the value of x^4 - x^2 + 6x - 4 when x=(1+√3 i)/2
x = (1 + √3 i)/2
x^2 = (1 - 3 + 2√3i) / 4 = (-1 + √3i)/2
x^4 = (1 - 3 - 2√3i)/4 = (-1 - √3i)/2
x^4 - x^2 + 6x - 4 = (-1 - √3i)/2 - (-1 + √3i)/2 + 6(1 + √3i)/2 - 4
= -1 + 2√3i