I have to factor the expression 64x^3-1
The x3 is the hint - it hints at the expression being the difference of two cubes, which can be factored into the form (ax - b)(a2x2 + abx + b2).
Since 64x3 = (4x)3 and 1 = 13, we can factor into
(4x - 1)(16x2 + 4x + 1)
Let's check ourselves by multiplying it out:
(4x)(16x2) + (4x)(4x) + (4x)(1) + (-1)(16x2) + (-1)(4x) + (-1)(1) =
64x3 + 16x2 + 4x -16x2 -4x + -1 = 64x3 - 1 (it worked!)
For reference, this form has a "cousin", the SUM of two cubes. It is factored in the form
(ax + b)(a2x2 - abx + b2).
So how do you know where to put the signs, since they look so much alike?
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.