Method

The FOIL Method

The FOIL method is a mnemonic for remembering how to multiply two binomials. The word FOIL is an acronym for the words

F irst,

O uter,

I nner,

and

L ast .

Consider, for example, the following product.

(x + 6) (3 x - 2)

These two binomials can be multiplied by following five steps.

Multiply the First Terms

Start by multiplying the first terms of each binomial. In this case, multiply

x

3 x .

(x + 6) (3 x - 2) = x (3 x) - 2 x + 18 x - 12

The empty box is there as a reminder that there are still missing terms.

Multiply the Outer Terms

Next, multiply the outer terms — that is, multiply the first term of the left-hand side binomial by the second term of the right-hand side binomial. In this case, multiply

x

- 2 .

(x + 6) (3 x - 2) = x (3 x) + x (- 2) 18 x - 12

Multiply the Inner Terms

Now multiply the inner terms — that is, multiply the second term of the left-hand side binomial by the first term of the right-hand side binomial. In this case, multiply

6

3 x .

(x + 6) (3 x - 2) = x (3 x) + x (- 2) + 6 (3 x) - 12

Multiply the Last Terms

Next, multiply the last terms of each binomial — that is, multiply the second term of the left-hand side binomial by the second term of the right-hand side binomial. In this case, multiply

6

- 2 .

(x + 6) (3 x - 2) = x (3 x) + x (- 2) + 6 (3 x) + 6 (- 2)

Simplify

Finally, find each product and combine like terms, if any, to simplify the resulting expression.

(x + 6) (3 x - 2) = x (3 x) + x (- 2) + 6 (3 x) + 6 (- 2)

CommutativePropMult

Commutative Property of Multiplication

(x + 6) (3 x - 2) = 3 x \cdot x + x (- 2) + 6 (3 x) + 6 (- 2)

ProdToPowTwoFac

$a \cdot a = a^{2}$

(x + 6) (3 x - 2) = 3 x^{2} + x (- 2) + 6 (3 x) + 6 (- 2)

MultPosNeg

$a (- b) = - a \cdot b$

(x + 6) (3 x - 2) = 3 x^{2} - x (2) + 6 (3 x) - 6 (2)

Multiply

(x + 6) (3 x - 2) = 3 x^{2} - 2 x + 18 x - 12

AddTerms

Add terms

(x + 6) (3 x - 2) = 3 x^{2} + 16 x - 12

The following applet illustrates the FOIL method using two arbitrary binomials.

Like any other polynomial multiplication, the FOIL method is based on the Distributive Property.

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