How to Factor a Sum of Cubes?

To factor a sum of cubes, use the following formula: a³ + b³ = (a + b)(a² - ab + b²).

Here's a breakdown of the process:

1. Identify 'a' and 'b': Recognize that you have an expression in the form of something cubed plus something else cubed (a³ + b³). Determine what 'a' and 'b' are by taking the cube root of each term.

2. Apply the Formula: Substitute 'a' and 'b' into the formula (a + b)(a² - ab + b²).

3. Simplify: Simplify the resulting expression.

Example:

Let's factor x³ + 8.

1. Identify 'a' and 'b':

   • a³ = x³ => a = x
   • b³ = 8 => b = 2 (since 2³ = 8)

2. Apply the Formula:

   • (a + b)(a² - ab + b²) = (x + 2)(x² - x(2) + 2²)

3. Simplify:

   • (x + 2)(x² - 2x + 4)

Therefore, x³ + 8 factors to (x + 2)(x² - 2x + 4).

Key Points:

• The quadratic factor (a² - ab + b²) resulting from factoring a sum of cubes usually cannot be factored further using real numbers.
• Remember the formula! It's the key to factoring sums of cubes.

Difference of Cubes Formula (for comparison):

It's helpful to remember the difference of cubes formula as well: a³ - b³ = (a - b)(a² + ab + b²). Notice the sign changes are consistent.