1. Special Products
What makes these products "special"?
The algebraic products on this page are used all the time in this chapter, and most of the math you will come across later. They are "special" because they are very common, and they're worth knowing.
If you can recognize these products easily, it makes your life easier later on.
Special Products involving Squares
The following special products come from multiplying out the brackets. You'll need these often, so it's worth knowing them well.
a(x + y) = ax + ay (Distibutive Law)
(x + y)(x − y) = x2 − y2 (Difference of 2 squares)
(x + y)2 = x2 + 2xy + y2 (Square of a sum)
(x − y)2 = x2 − 2xy + y2 (Square of a difference)
Examples using the special products
Example 1: Multiply out 2x(a − 3)
Example 2: Multiply `(7s + 2t)(7s − 2t)`
Example 3: Multiply (12 + 5ab)(12 − 5ab)
Example 4: Expand (5a + 2b)2
Example 5: Expand (q − 6)2
Example 6: Expand (8x − y)(3x + 4y)
Example 7: Expand (x + 2 + 3y)2
Special Products involving Cubes
The following products are just the result of multiplying out the brackets.
(x + y)3 = x3 + 3x2y + 3xy2 + y3 (Cube of a sum)
(x − y)3 = x3 − 3x2y + 3xy2 − y3 (Cube of a difference)
(x + y)(x2 − xy + y2) = x3 + y3 (Sum of 2 cubes)
(x − y)(x2 + xy + y2) = x3 − y3 (Difference of 2 cubes)
These are also worth knowing well enough so you recognize the form, and the differences between each of them. (Why? Because it's easier than multiplying out the brackets and it helps us solve more complex algebra problems later.)
Example: Expand `(2s + 3)^3`
(1) (s + 2t)(s − 2t)
(2) (i1 + 3)2
(3) (3x + 10y)2
(4) (3p − 4q)2