This lesson covers factorization of expressions containing two or more variables and cubic expressions.
Organizing with Respect to One Variable
For an expression containing multiple variables, organizing it with respect to the variable of the lowest degree makes its structure easier to see.
Example ①
Factor x² + xy − 2x − 2y.
Answer:Organize with respect to y: (x − 2)y + (x² − 2x) = (x − 2)y + x(x − 2) = (x − 2)(x + y)
Factor x² + xy − 2x − 2y.
Answer:Organize with respect to y: (x − 2)y + (x² − 2x) = (x − 2)y + x(x − 2) = (x − 2)(x + y)
Cubic Formulas
- a³ + b³ = (a + b)(a² − ab + b²)
- a³ − b³ = (a − b)(a² + ab + b²)
Example ②
(1) x³ + 8 (2) 27x³ − 1
Answer:
(1) x³ + 2³ = (x + 2)(x² − 2x + 4)
(2) (3x)³ − 1³ = (3x − 1)(9x² + 3x + 1)
(1) x³ + 8 (2) 27x³ − 1
Answer:
(1) x³ + 2³ = (x + 2)(x² − 2x + 4)
(2) (3x)³ − 1³ = (3x − 1)(9x² + 3x + 1)
Key Points
- Organize an expression with multiple variables with respect to one variable
- Apply the formula a³ ± b³ to sums and differences of cubes
- In the squared factor, use a minus sign for a sum and a plus sign for a difference
Practice Problems
- Factor ab − a − b + 1.
- Factor x³ − 27.
- Factor 8x³ + 125.
Answers and Explanations
- Answer:(a − 1)(b − 1)
Explanation: a(b − 1) − (b − 1) = (b − 1)(a − 1). - Answer:(x − 3)(x² + 3x + 9)
Explanation: Use the formula for a³ − b³. - Answer:(2x + 5)(4x² − 10x + 25)
Explanation: Apply the formula to (2x)³ + 5³.