Product formulas

A product is a number or a quantity obtained by multiplying two or more numbers together.

1. ( a – b )²= a² – 2ab + b²

Example:

Expand: (x - 4 y)²

Solution:

(x - 4 y)² = x² - 2(x)(4 y) + (4 y)²

= x² - 8xy + 16 y²

2. ( a + b )² = a² + 2ab + b²

Example:

Expand: (x + 2y)²

Solution:

(x + 2 y)² = x² + 2(x)(2 y) + (2 y)²

= x² + 4xy + 4 y²

3. ( a – b )³ = a³ – 3a²b + 3ab² – b³

Example:

Expand: (4x - 3y)³

Solution:

(4x - 3y)³ = (4x)³ - 3(4x)² (3y) + 3(4x)(3y)² - (3y)³

= 64x³ - 144x² y + 108xy ² - 27 y³

4. ( a + b )³ = a³ + 3a²b + 3ab² + b³

Example:

Expand: (2x + 3)³

Solution:

(2x + 3)³ = (2x)³ + 3(2x)² (3) + 3(2x)(3)² + (3)³

= 8x³ + 36x² + 54x + 27

5. ( a – b )⁴ = a⁴ – 4a³b + 6a²b² – 4ab³ + b⁴

6. ( a + b )⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

7. ( a + b + c )² = a² + b² + c² + 2ab + 2ac + 2bc

8. ( a + b + c+ . . . . )² = a² + b² + c² + . . . . + 2(ab + ac + bc + . . . .)