Factoring Sums and Differences

Definition: When you learn to factor quadratics, there are three other formulas that are usually introduced at the same time.  

Difference of two squares:

a² – b² = (a+ b)(a – b)

Sum of two cubes:

a³ + b³ = (a + b)(a² – ab + b²)

Difference of two cubes:

a³ – b³ = (a – b)(a² + ab + b²)

Memorizing these formulas will help you solve quadratic equations quickly.

Example:Factor using the best method: 27x³ – 64b³

Solution: 

Step 1: Determine the best method of factoring. Since 27 and 64 are both perfect cubes, we can write this as:

(3x)³ – (4y)³

We can now plug these values into the difference of two cubes formula, a³ – b³ = (a – b)(a² + ab + b²) where a = 3x and b = 4y.

Step 2: Plugging these values into the difference of two cubes formula give us:

(3x)³ – (4y)³ = (3x – 4y)[(3x)² + 3x*4y + (4y)²]

Step 3: Simplify to finish:

(3x – 4y) [(3x)² + 3x*4y + (4y)²] = (3x – 4y)(9x² + 12xy + 16y²)

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