Factor 64x^3-1

Question

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

Answer 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?

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.

Answer 2

You have to use the sum and difference of cubes formulas:

a³ + b3 = (a + b)(a² – ab + b²)
a³ – b³ = (a – b)(a² + ab + b²)

a³ ± b³ = (a [same sign] b)(a² [opposite sign] ab [always positive] b²)

The pattern is easily memorized by using the mnemonic SOAP. S stands for Same (the first sign is the same as the binomial being factored),
O stands for Opposite (the second sign is the opposite of the binomial being factored), and AP stands for Always Positive (the last sign is always positive.)

64x³-1 = (4x)³-1³ = (4x -1)((4x)²+(4x)(1)+(1)²) = (4x - 1)(16x²+4x+1)