While the FOIL method discussed in
section 10.2 is always an option, certain types pairs are worth remembering as they allow you to expand them without
much effort (and will prove useful later when factoring). The first such pair is the "product of a sum and difference", or (a + b)(a - b)
(a + b)(a - b)
a*a + a*-b + b*a + b*-b
a
2 - ab + ab - b
2
a
2 - b
2
As you can see, when you multiply a sum and difference, the "middle" term cancels out leaving just a difference of squares. The second pair is when you are squaring a binomial, so
(a + b)
2 or (a - b)
2
(a + b)
2
(a + b)(a + b)
a*a + ab + ba + b*b
a
2 + 2ab + b
2
When you square a term you get the sum of each term squares and twice the product of the terms. Likewise (a - b)
2 = a
2 - 2ab + b
2.
Knowing these can help with mental math as well, for example you can think of 18 * 22 as (20 - 2)*(20 + 2) = 20
2 - 2
2 = 400 - 4 = 396. Or you can think of 23
2 as
(20 + 3)
2 = 20
2 + 2*(20*3) + 3
2 = 400 + 120 + 9 = 529.