What is the Formula for Cube Factor?

The "cube factor" refers to the formulas used to factor the sum or difference of two cubes. Here's a breakdown:

Formulas:

• Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)
• Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)

Explanation:

These formulas allow you to break down expressions in the form of a³ + b³ or a³ - b³ into simpler factors. Understanding these formulas is crucial for simplifying algebraic expressions and solving equations.

How to Use Them:

1. Identify 'a' and 'b': Determine what values, when cubed, give you the terms in your expression. For example, in x³ + 8, 'a' is x (because x³ = x³) and 'b' is 2 (because 2³ = 8).

2. Choose the correct formula: Decide whether you have a sum of cubes (a³ + b³) or a difference of cubes (a³ - b³).

3. Substitute 'a' and 'b' into the formula: Replace 'a' and 'b' in the appropriate formula with the values you identified.

4. Simplify: Simplify the resulting expression if possible.

Examples:

• Factor x³ + 8:

  • a = x, b = 2
  • Using the sum of cubes formula: (x + 2)(x² - x*2 + 2²)
  • Simplified: (x + 2)(x² - 2x + 4)

• Factor 27y³ - 1:

  • a = 3y (because (3y)³ = 27y³), b = 1
  • Using the difference of cubes formula: (3y - 1)((3y)² + (3y)*1 + 1²)
  • Simplified: (3y - 1)(9y² + 3y + 1)

Mnemonic Device:

A helpful mnemonic device to remember the signs in the formulas is "SOAP":

• Same: The first sign in the factored form is the same as the sign in the original expression.
• Opposite: The second sign is the opposite of the sign in the original expression.
• Always Positive: The last sign is always positive.

So, for a³ + b³: (a + b)(a² - ab + b²) (Same, Opposite, Always Positive)
And for a³ - b³: (a - b)(a² + ab + b²) (Same, Opposite, Always Positive)