1. (a) Simplify \( \frac{3x^3 - x - 2}{x^2 - 1} \), (b) Hence, or otherwise, express \( \frac{3x^3 - x - 2}{x^2 - 1} \) as a single fraction in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2006 - Paper 4

First, we will factor the denominator, which is a difference of squares:

$x^2 - 1 = (x - 1)(x + 1)$

Next, we aim to factor the numerator, (3x^3 - x - 2). To do this, we can apply polynomial long division or synthetic division, but initially, we will test for rational roots to find any factors. We find that:

• Using synthetic division with (x = 1):
  • The polynomial reduces nicely, so we have:

$3x^3 - x - 2 = (x - 1)(3x^2 + 3x + 2)$

This allows us to simplify our expression to:

$\frac{(x - 1)(3x^2 + 3x + 2)}{(x - 1)(x + 1)}$

Cancelling the (x - 1) terms gives:

$\frac{3x^2 + 3x + 2}{x + 1}$