Unit 6 - Objective 1 - Special Products

Some special products you will see:

 common factor a (x + y) = ax + ay difference of two squares (x + y)(x - y) = x2 - y2 perfect square (x + y)2 = (x + y)(x + y) = x2 + 2xy + y2 " (x - y)2 = (x - y)(x - y) = x2 - 2xy + y2 general product (x + a)(x + b) = x2 + (a + b) x + ab perfect cube (x + y)3 = (x + y)(x + y)(x + y) = x3 + 3x2y + 3xy2 + y3 " (x - y)3 = (x - y)(x - y)(x - y) = x3 - 3x2y + 3xy2 - y3 sum of two cubes (x + y)(x2 - xy + y2) = x3 + y3 difference of two cubes (x - y)(x2 + xy + y2) = x3 - y3

Examples:

1.   (use distributive property)

2. (x + 6) (x - 6)   (difference of two squares)
= (x)2 - (6)2
= x2 - 36

3. 3x (3x + 5y) (3x - 5y)
= 3x ((3x)2 - (5y)2)

(Multiply the two binomials first because they are
the difference of two squares)

(Now the distributive property)

4. (5a + 2b)2   (perfect square)

= 25a2 + 10ab + 10ab + 4b2   (combine similar terms)
= 25a2 + 20ab + 4b2

or

using the perfect square formula this is the same as

 (5a + 2b)2 = (5a)2 + 2 (5a) (2b) + (2b)2 = 25a2 + 20ab + 4b2

5.

6. (x + 2)3   (perfect cube)
(you could multiply the first 2 quantities)
= (x2 + 2x + 2x + 4) (x + 2)   (combine like terms)
= (x2 + 4x + 4) (x + 2)
= (x2) (x2) + (x2) (2) + (4x) (x) + (4x) (2) + (4) (x) + (4) (2)
= x3 + 2x2 + 4x2 + 8x + 4x + 8  (combine similar terms)
= x3 + 6x2 + 12x + 8