Question

x square minus 16 whole by x power minus 4​

Answer 1

What is the best way to factor x⁴-16?

The difference of squares: a² - b² factors to (a - b) (a + b)

Let x⁴ = (x²)²

Let 16 = 4²

Now by difference of squares x⁴ - 16 factors as (x² - 4)(x² + 4)

The front factor, (x² - 4), can factor by difference of squares again as (x - 2)(x + 2)

In this way x⁴ - 16 = (x - 2)(x + 2)(x² + 4)

Only if you dip into imaginary numbers can you factor this any farther.

If you use imaginary numbers then (x² + 4) can become (x² -(-4)) and the square root of -4 is 2i

so (x² - (-4)) becomes (x + 2i)(x - 2i). As a result using imaginary numbers, you can factor x⁴ - 16 into even smaller factors

x⁴ - 16 = (x - 2)(x + 2)(x - 2i)(x + 2i)