Kevin S. answered • 01/14/13

The x^{3} is the hint - it hints at the expression being the difference of two cubes, which can be factored into the form (ax - b)(a^{2}x^{2} + abx + b^{2}).

Since 64x^{3} = (4x)^{3} and 1 = 1^{3}, we can factor into

(4x - 1)(16x^{2} + 4x + 1)

Let's check ourselves by multiplying it out:

(4x)(16x^{2}) + (4x)(4x) + (4x)(1) + (-1)(16x^{2}) + (-1)(4x) + (-1)(1) =

64x^{3} + 16x^{2} + 4x -16x^{2} -4x + -1 = 64x^{3} - 1 (it worked!)

For reference, this form has a "cousin", the SUM of two cubes. It is factored in the form

(ax + b)(a^{2}x^{2} - abx + b^{2}).

So how do you know where to put the signs, since they look so much alike?

Remember this:

1. The sign between the cubes (+ or -) goes between the terms in the LINEAR factor: ax and b.

2. The OTHER sign goes in the quadratic factor, in between the first and second terms.

3. The sign between the second and third term will be positive.

Kevin S.

Elana - I haven't heard that mnemonic before. I'll have to remember it... Thanks!01/14/13