The sum or difference of two cubes can be factored into a product of a binomial and a trinomial using specific formulas. Let's break down how to do it.
Formulas for Sum and Difference of Cubes
Here are the general formulas:
- Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)
- Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)
Understanding the Formulas
Notice the patterns in these formulas:
- Sign in the Binomial: The sign in the binomial factor (the first factor) is the same as the sign in the original expression.
- Sign in the Trinomial: The sign of the middle term in the trinomial factor (the second factor) is the opposite of the sign in the original expression. The last term is always positive.
- No '2' in the Trinomial: There is no "2ab" term in the trinomial, unlike the expansion of (a+b)² or (a-b)².
Steps for Factoring
- Identify 'a' and 'b': Determine what is being cubed in each term of the expression. For example, in x³ + 8, 'a' is x and 'b' is 2 (since 2³ = 8).
- Apply the Correct Formula: Use either the sum of cubes formula or the difference of cubes formula, depending on the expression.
- Substitute 'a' and 'b': Plug the values of 'a' and 'b' you identified into the formula.
- Simplify: Simplify the resulting expression.
Examples
Example 1: Factoring x³ + 8 (Sum of Cubes)
- a = x, b = 2
- Formula: a³ + b³ = (a + b)(a² - ab + b²)
- Substitute: x³ + 8 = (x + 2)(x² - x(2) + 2²)
- Simplify: x³ + 8 = (x + 2)(x² - 2x + 4)
Example 2: Factoring 27y³ - 1 (Difference of Cubes)
- a = 3y (since (3y)³ = 27y³), b = 1
- Formula: a³ - b³ = (a - b)(a² + ab + b²)
- Substitute: 27y³ - 1 = (3y - 1)((3y)² + (3y)(1) + 1²)
- Simplify: 27y³ - 1 = (3y - 1)(9y² + 3y + 1)
Common Mistakes to Avoid
- Incorrect Signs: Double-check the signs in the binomial and trinomial factors. Remember the "same-opposite-always positive" rule.
- Forgetting to Square 'a' and 'b': Make sure you square 'a' and 'b' correctly when substituting into the trinomial factor.
- Including a '2' in the Trinomial: Avoid mistakenly adding a '2ab' term, as there is none in these formulas.
- Not identifying 'a' and 'b' correctly: Ensure you correctly identify the terms being cubed. Don't forget about coefficients!
Factoring the sum and difference of cubes relies on memorizing (or understanding) the two key formulas and then carefully substituting and simplifying. Practice makes perfect!
