The basic method of factorization is to use expansion formulas in reverse.
Factorization Formulas
- a² + 2ab + b² = (a + b)²
- a² − 2ab + b² = (a − b)²
- a² − b² = (a + b)(a − b)
- x² + (a + b)x + ab = (x + a)(x + b)
Example ①
(1) x² + 10x + 25 (2) x² − 12x + 36 (3) x² − 49
Answer:(1) (x + 5)² (2) (x − 6)² (3) (x + 7)(x − 7)
(1) x² + 10x + 25 (2) x² − 12x + 36 (3) x² − 49
Answer:(1) (x + 5)² (2) (x − 6)² (3) (x + 7)(x − 7)
Form x² + (a+b)x + ab
Find two numbers a and b whose product is the constant term and whose sum is the coefficient of the first-degree term.
Example ②
(1) x² + 7x + 12 (2) x² − x − 12
Answer:
(1) Product 12, sum 7 → 3 and 4 → (x + 3)(x + 4)
(2) Product −12, sum −1 → 3 and −4 → (x + 3)(x − 4)
(1) x² + 7x + 12 (2) x² − x − 12
Answer:
(1) Product 12, sum 7 → 3 and 4 → (x + 3)(x + 4)
(2) Product −12, sum −1 → 3 and −4 → (x + 3)(x − 4)
Key Points
- For a perfect square, check whether the middle term is ±2ab
- Recognize the form a² − b² as a difference of squares
- Find the two numbers whose product is the constant term and whose sum is the coefficient of the first-degree term
Practice Problems
- Factor x² + 14x + 49.
- Factor x² − 81.
- Factor x² + 2x − 15.
Answers and Explanations
- Answer:(x + 7)²
Explanation: 49=7² and 14=2·7. It is a perfect square. - Answer:(x + 9)(x − 9)
Explanation: Difference of squares. - Answer:(x + 5)(x − 3)
Explanation: Product −15, sum 2 → 5 and −3.