I have to factor the expression 64x^3-1

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.

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