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# The FOIL Method

Consider how we find the product of two binomials x + 3 and x + 5 using the distributive property twice:

 (x + 3)(x + 5) = (x + 3)x + (x + 3)5 Distributive property = x2 + 3x + 5x + 15 Distributive property = x2 + 8x + 15 Combine like terms.

There are four terms in the product. The term x2 is the product of the first term of each binomial. The term 5x is the product of the two outer terms, 5 and x. The term 3x is the product of the two inner terms, 3 and x. The term 15 is the product of the last two terms in each binomial, 3 and 5. It may be helpful to connect the terms multiplied by lines. F = First termsO = Outer terms I = Inner terms L = Last terms

So instead of writing out all of the steps in using the distributive property, we can get the result by finding the products of the first, outer, inner, and last terms. This method is called the FOIL method.

For example, letâ€™s apply FOIL to the product (x - 3)(x + 4): If the outer and inner products are like terms, you can save a step by writing down only their sum.

The product of two binomials always has four terms before combining like terms. The product of two trinomials always has nine terms before combining like terms. How many terms are there in the product of a binomial and trinomial?

Example 1

Multiplying binomials

Use FOIL to find the products of the binomials.

a) (2x - 3)(3x + 4)

b) (2x3 + 5)(2x3 - 5)

c) (m + w)(2m - w)

d) (a + b)(a - 3)

Solution

a) (2x - 3)(3x + 4) b) (2x3 + 5)(2x3 - 5) = 4x6 - 10x3 + 10x3 - 25 = 4x6 - 25

c) (m + w)(2m - w) = 2m2 - mw +2mw - w2 = 2m2 + mw - w2

d) (a + b)(a - 3) = a2 - 3a + ab - 3b There are no like terms.