The sum-and-product pattern
Identity for (x + a)(x + b)
A hidden pattern
We know that (x + 3)(x + 4) = x² + 7x + 12. Also (x + 6)(x + 7) = x² + 13x + 42. Look at the numbers 3 and 4, then at 7 and 12.
How do 3 and 4 give the 7 and the 12?
What this lesson covers
The idea
(x + a)(x + b) = x² + (a + b)x + ab.
A hidden pattern
We know that (x + 3)(x + 4) = x² + 7x + 12. Also (x + 6)(x + 7) = x² + 13x + 42. Look at the numbers 3 and 4, then at 7 and 12.
How do 3 and 4 give the 7 and the 12?
- 7 = 3 + 4 and 12 = 3 × 4
- 7 = 3 × 4 and 12 = 3 + 4
- There is no link
Count the tiles
Change a and b in the rectangle with sides x + a and x + b. Count the x-tiles and the unit tiles each time. Try three different pairs.
Sum and product
(x + a)(x + b) = x² + (a + b)x + ab. The number of x-tiles is the sum a + b. The number of unit tiles is the product ab.
Check: (x + 6)(x + 7) = x² + 13x + 42, because 6 + 7 = 13 and 6 × 7 = 42.
Notes
(x + a)(x + b) = x² + (a + b)x + ab: the x-term is the sum, the constant term is the product.
Check yourself
(x + 5)(x + 2) = x² + ?x + 10. What number goes in the box?
Answer: 7
5 + 2 = 7, and 5 × 2 = 10 is the constant.
(x + 8)(x + 3) = x² + 11x + ?. What is the constant term?
Answer: 24
8 × 3 = 24, and 8 + 3 = 11 is the x-term.
Which is (x + 4)(x + 9)?
Which is (x − 2)(x + 5)? Take a = −2 and b = 5.
- x² + 36x + 13. Sum and product are swapped. The x-term is the sum 13 and the constant is the product 36.
- x² + 13x + 13. The constant term is the product 4 × 9 = 36, not the sum.
- x² + 13x + 36 — correct. Yes: 4 + 9 = 13 and 4 × 9 = 36.
- x² + 3x − 10 — correct. Yes: −2 + 5 = 3 and (−2) × 5 = −10.
- x² + 7x + 10. a is −2, not 2. The sum is −2 + 5 = 3 and the product is −10.
- x² + 3x + 10. The product (−2) × 5 is negative: −10.