Learn how to factor expressions in the form x²＋bx＋c into (x＋p)(x＋q).
Factoring x²＋bx＋c
Find p and q such that p×q＝c and p＋q＝b.
Example 1
Factor x²＋7x＋12.
Solution:
Two numbers whose product is 12 and whose sum is 7: 3 and 4
x²＋7x＋12 ＝ (x＋3)(x＋4)
Factor x²＋7x＋12.
Solution:
Two numbers whose product is 12 and whose sum is 7: 3 and 4
x²＋7x＋12 ＝ (x＋3)(x＋4)
Example 2 (Pay attention to the signs)
Factor x²－5x＋6.
Solution:
Two numbers whose product is 6 and whose sum is －5: －2 and －3
x²－5x＋6 ＝ (x－2)(x－3)
Factor x²－5x＋6.
Solution:
Two numbers whose product is 6 and whose sum is －5: －2 and －3
x²－5x＋6 ＝ (x－2)(x－3)
Example 3 (Expressions in the form ax²)
Factor 2x²＋7x＋3.
Solution (cross-multiplication method):
Check (2x＋1)(x＋3): 2×3＋1×1＝7 ✓
(2x＋1)(x＋3)
Factor 2x²＋7x＋3.
Solution (cross-multiplication method):
Check (2x＋1)(x＋3): 2×3＋1×1＝7 ✓
(2x＋1)(x＋3)
Key Points
- Find two numbers whose “product is c and sum is b.”
- Pay attention to combinations of signs (++ gives a sum, +- gives a difference, and -- gives a negative sum.)
- For expressions in the form ax², systematically search using the cross-multiplication method.
Practice Problems
- Factor x²＋3x－10.
- Factor x²－8x＋15.
Answers and Explanations
- Answer:(x＋5)(x－2)
Explanation: Two numbers whose product is －10 and whose sum is 3: 5 and －2. x²＋3x－10＝(x＋5)(x－2). Check: (x+5)(x-2)＝x²-2x+5x-10＝x²+3x-10 ✓ - Answer:(x－3)(x－5)
Explanation: Two numbers whose product is 15 and whose sum is －8: －3 and －5. x²－8x＋15＝(x－3)(x－5). Check: (x-3)(x-5)＝x²-5x-3x+15＝x²-8x+15 ✓