Question

Factorise the following:

x⁴ – 14x² + 1

Answer 1

Given: x⁴ – 14x² + 1

Step-wise calculation:

1. Expand the given binomials:

(x² + 4x + 1)(x² – 4x + 1)

Multiply out using the distributive property:

= x²(x² – 4x + 1) + 4x(x² – 4x + 1) + 1(x² – 4x + 1)

= x⁴ – 4x³ + x² + 4x³ – 16x² + 4x + x² – 4x + 1

2. Simplify by combining like terms:

The –4x³ and +4x³ cancel.

Combine the x² terms:

x² – 16x² + x² = (1 – 16 + 1)x²

x² – 16x² + x² = –14x²

Combine the x terms: 4x – 4x = 0

So, what remains is x⁴ – 14x² + 1

(x² + 4x + 1)(x² – 4x + 1) = x⁴ – 14x² + 1