Question

Factorise:

3x³ – x² – 3x + 1

Answer 1

Let p(x) = 3x³ – x² – 3x + 1

Constant term of p(x) = 1

Factor of 1 are ±1

By trial, we find that p(1) = 0, so (x – 1) is a factor of p(x)

Now, we see that 3x³ – x² – 3x + 1

= 3x³ – 3x² + 2x² – 2x – x + 1

= 3x²(x – 1) + 2x(x – 1) – 1(x – 1)

= (x – 1)(3x² + 2x – 1)

Now, (3x² + 2x – 1) = 3x² + 3x – x – 1  ...[By splitting middle term]

= 3x(x + 1) – 1(x + 1)

= (x + 1)(3x – 1)

∴ 3x³ – x² – 3x + 1 = (x – 1)(x + 1)(3x – 1)