“Cross multiplication” is a technique used when the coefficient of the quadratic term is not 1.
The Idea of Cross Multiplication
When factoring ax² + bx + c into (px + q)(rx + s), find numbers satisfying pr = a, qs = c, and ps + qr = b.
Write them with × marks as shown in the diagram, and adjust them so that the sum of the diagonal products equals the middle-term coefficient.
p ＼ q r ／ s ps + qr = b
Example ①
Factor 2x² + 7x + 3.
Answer:2 = 2×1 and 3 = 1×3. For 2x²+7x+3, 2·3 + 1·1 = 7 ✓.
Therefore, (2x + 1)(x + 3)
Factor 2x² + 7x + 3.
Answer:2 = 2×1 and 3 = 1×3. For 2x²+7x+3, 2·3 + 1·1 = 7 ✓.
Therefore, (2x + 1)(x + 3)
Example ②
Factor 6x² − x − 12.
Answer:6 = 2×3 and −12 = (+3)·(−4). 2·(−4) + 3·3 = −8 + 9 = 1.
Reverse the signs to obtain the middle term −x: 2·4 + 3·(−3) = 8 − 9 = −1 ✓.
Therefore, (2x − 3)(3x + 4)
Factor 6x² − x − 12.
Answer:6 = 2×3 and −12 = (+3)·(−4). 2·(−4) + 3·3 = −8 + 9 = 1.
Reverse the signs to obtain the middle term −x: 2·4 + 3·(−3) = 8 − 9 = −1 ✓.
Therefore, (2x − 3)(3x + 4)
Key Points
- Try factoring the quadratic coefficient and constant term to match the middle term
- The signs are determined by the signs of a > 0, c, and b
- Use a × diagram to check your work until you become familiar with the method
Practice Problems
- Factor 3x² + 8x + 4.
- Factor 2x² − 7x + 3.
- Factor 6x² + 5x − 6.
Answers and Explanations
- Answer:(3x + 2)(x + 2)
Explanation: 3·2 + 1·2 = 8 ✓. - Answer:(2x − 1)(x − 3)
Explanation: 2·(−3) + 1·(−1) = −7 ✓. - Answer:(2x + 3)(3x − 2)
Explanation: 2·(−2) + 3·3 = 5 ✓.