Question

Factorise:

x⁶ – 7x³ – 8

Answer 1

Factorising x⁶ – 7x³ – 8 we get,

x⁶ – 7x³ – 8 = x⁶ – 8x³ + x³ – 8

= x³(x³ – 8) + 1(x³ – 8)

= (x³ – 8) (x³ + 1)

= [(x)³ – (2)³] [(x)³ + 1³]

We know that,

a³ + b³ = (a + b) (a² – ab + b²)

a³ – b³ = (a – b) (a² + ab + b²)

∴ x³ + 1³ = (x + 1) (x² – x + 1)

And,

(x)³ – (2)³ = (x – 2) (x² + 2x + 4)

∴ [(x)³ – (2)³] [(x)³ + 1³] = (x – 2) (x² + 2x + 4) (x + 1) (x² – x + 1).

Hence, x⁶ – 7x³ – 8 = (x – 2) (x² + 2x + 4) (x + 1) (x² – x + 1).