Formula 3
Use the expansion formula (x+a)(x+b) = x² + (a+b)x + ab in reverse. Focus on the product of the constant terms and the sum of the coefficient of x, and find the two numbers a, b.
Cross-multiplication formula
x² + (a+b)x + ab = (x + a)(x + b)
Find two numbers a, b whose product is the constant term and whose sum is the coefficient of x.
📘 Example ①
Factor x² + 5x + 6.
Answer: The two numbers whose product is 6 and whose sum is 5 are 2 and 3, so (x + 2)(x + 3)
Factor x² + 5x + 6.
Answer: The two numbers whose product is 6 and whose sum is 5 are 2 and 3, so (x + 2)(x + 3)
📘 Example ②
Factor x² − 3x − 10.
Answer: The two numbers whose product is −10 and whose sum is −3 are 2 and −5, so (x + 2)(x − 5)
Factor x² − 3x − 10.
Answer: The two numbers whose product is −10 and whose sum is −3 are 2 and −5, so (x + 2)(x − 5)
💡 Key Points
- List the factor pairs of the constant term and check their sum
- Pay attention to the signs (when the constant term is negative, the two numbers have different signs)
- Develop the habit of expanding after factoring to check your answer
Practice Problems
- Factor x² + 7x + 12.
- Factor x² − x − 6.
Answers and Explanations
- Answer: The two numbers whose product is 12 and whose sum is 7 are 3 and 4, so (x + 3)(x + 4)
- Answer: The two numbers whose product is −6 and whose sum is −1 are 2 and −3, so (x + 2)(x − 3)