1. Algebra 1
Section 9.5

2. Special Products
Recognizing the patterns in special binomial products can save you time.
The first special product is the square of a sum.

3. Special Products
Square of a sum:
(a + b)2 = a2 + 2ab + b2

4. Example 2 (x + 2)2 = x2 + 2(x)(2) + 22 = x2 + 4x + 4 (3x + 4)2 =

5. Special Products
Square of a difference:
(a – b)2 = a2 – 2ab + b2

6. Example 3 (y – 3)2 = y2 – 2(y)(3) + 32 = y2 – 6y + 9 (2x – 5y)2 =
(2x)2 – 2(2x)(5y) + (5y)2
= 4x2 – 20xy + 25y2

7. Special Products
Two binomial expressions, one indicating the sum of two terms, and the other indicating the difference of the same terms, are called conjugates.

8. A product of conjugates is the difference of two squares.
Example 4
Multiply (a + b)(a – b). a2
– ab
+ ab
– b2
a2 – b2
A product of conjugates is the difference of two squares.

9. Example 5 (x – 4)(x + 4) = x2 – 42 = x2 – 16 (3x + 7y)(3x – 7y) =

10. Special Products Square of a sum: (a + b)2 = a2 + 2ab + b2
Square of a difference:
(a – b)2 = a2 – 2ab + b2
Product of conjugates:
(a + b)(a – b) = a2 – b2

11. Example 6 87 × 93 = (90 – 3)(90 + 3) = 902 – 32 = 8100 – 9 = 8091
722 = (70 + 2)2
= (70)(2) + 22 =
= 5184

12. Example 7 Expand (a – b)3. (a – b)3 = (a – b)(a – b)2 + 2ab2 – b3
= (a – b)(a2 – 2ab + b2)
= a3 – 2a2b + ab2 – a2b
+ 2ab2 – b3
= a3 – 3a2b + 3ab2 – b3

13. Homework: pp