4.6 Special Products
Learning Objectives
After completing this section, you should be able to:
- Recognize and expand squares of binomials using patterns.
- Recognize and expand products of conjugates (difference of squares).
- Verify these patterns using the Distributive Property.
- Simplify and write results in standard form.
Squares of Binomials
When a binomial is squared, it is multiplied by itself:
(a + b)² = (a + b)(a + b)
Using the Distributive Property:
a² + 2ab + b²
Similarly,
(a − b)² = a² − 2ab + b²
Patterns
| Binomial | Expanded Form | Description |
|---|---|---|
| (a + b)² | a² + 2ab + b² | Square of a sum |
| (a − b)² | a² − 2ab + b² | Square of a difference |
Example 4.6.1
Expand (x + 5)²
Solution
x² + 2(5)(x) + 5²
= x² + 10x + 25
Alternate method (FOIL):
(x + 5)(x + 5) = x² + 5x + 5x + 25 = x² + 10x + 25
Example 4.6.2
Expand (y − 3)²
Solution
y² − 2(3)(y) + 3²
= y² − 6y + 9
Example 4.6.3
Expand (2a + 7)²
Solution
(2a)² + 2(2a)(7) + 7²
= 4a² + 28a + 49
Example 4.6.4
Expand (3m − 4)²
Solution
(3m)² − 2(3m)(4) + 4²
= 9m² − 24m + 16
Example 4.6.5
Expand (x + y)²
Solution
x² + 2xy + y²
Try It
Product of Conjugates (Difference of Squares)
Two binomials that differ only by the sign between terms are called conjugates:
(a + b) and (a − b)
Their product is a difference of squares:
(a + b)(a − b) = a² − b²
Pattern:
(a + b)(a − b) = a² − b²
Note:
You may always verify this pattern using FOIL to see how the middle terms cancel.
Example 4.6.6
Simplify (x + 5)(x − 5)
Solution
x² − 25
FOIL check:
x² + 5x − 5x − 25 = x² − 25
Example 4.6.7
Simplify (2x + 5)(2x − 5)
Solution
(2x)² − 5² = 4x² − 25
Example 4.6.8
Simplify (a + b)(a − b)
Solution
a² − b²
Example 4.6.9
Simplify (3x + 4y)(3x − 4y)
Solution
9x² − 16y²
Example 4.6.10
Simplify (m + 4)(m − 4)
Solution
m² − 16
Try It
Summary of Special Product Patterns
| Expression | Expanded Form | Name |
|---|---|---|
| (a + b)² | a² + 2ab + b² | Square of a sum |
| (a − b)² | a² − 2ab + b² | Square of a difference |
| (a + b)(a − b) | a² − b² | Difference of squares |
Key Takeaways
- Binomial squares and conjugates follow predictable patterns.
- The middle terms in a difference of squares always cancel.
- Patterns speed up simplification, but FOIL always works.
- Write final answers in standard form.
