Factoring Sum and Difference of Cubes

Understanding the Formulas and Perfect Cubes00:00:00

This video teaches how to factor the sum and difference of two cubes. It's crucial to know the formulas and common perfect cubes like 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000. The formulas involve finding the cube roots (a and b values) and substituting them. A helpful acronym for remembering the signs in the formulas is "SOAP," which stands for "Same, Opposite, Always Positive."

Example 1: Factoring x³ - 2700:01:28

For the expression x³ - 27, the cube root of x³ is x (our 'a' value), and the cube root of 27 is 3 (our 'b' value). Since it's a difference of two cubes, we use the formula (a - b)(a² + ab + b²). This results in (x - 3)(x² + 3x + 9). The trinomial cannot be factored further.

Example 2: Factoring 8y³ + 100:02:30

In this example, 8y³ + 1, the cube root of 8y³ is 2y (our 'a' value), and the cube root of 1 is 1 (our 'b' value). This is a sum of two cubes, so we use the formula (a + b)(a² - ab + b²). Applying the SOAP acronym for signs, the factored form is (2y + 1)(4y² - 2y + 1).

Example 3: Factoring 64d³ - 12500:03:32

For 64d³ - 125, the cube root of 64d³ is 4d (our 'a' value), and the cube root of 125 is 5 (our 'b' value). Following the difference of cubes formula and the SOAP acronym, the expression factors to (4d - 5)(16d² + 20d + 25).

Example 4: Factoring 216c³ + 1000d³00:04:21

The final example is 216c³ + 1000d³. The cube root of 216c³ is 6c (our 'a' value), and the cube root of 1000d³ is 10d (our 'b' value). Applying the sum of cubes formula with the SOAP pattern, the factored expression is (6c + 10d)(36c² - 60cd + 100d²).