Question

Factor completely 16x^8 − 1.

Answer 1

(2x² - 1)(2x² + 1)(4
`x^(4)` + 1)

Explanation:

16
`x^(8)` - 1 ← is a difference of squares and factors in general as

a² - b² = (a - b)(a + b), thus

16
`x^(8)` - 1

= (4
`x^(4)` )² - 1²

= (4
`x^(4)` - 1)(4
`x^(4)` + 1) ← 4x
`x^(4)` - 1 is also a difference of squares

4
`x^(4)` - 1

= (2x² )² - 1²

= (2x² - 1)(2x² + 1)

Thus

16
`x^(8)` - 1 = (2x² - 1)(2x² + 1)(4
`x^(4)` + 1)

Answer 2

(4x^4 + 1) (2x^2 + 1) (2x^2 - 1)

Explanation: