# Question 486ad

Oct 19, 2016

FOIL provides a memory aid to help students learning to multiply binomials.

#### Explanation:

The FOIL method provides a mnemonic to help algebra students remember how to multiply binomials. While most students are fine with multiplying a monomial by a binomial and distributing-

$a \left(b + c\right) = a b + a c$

-they often have difficulty generalizing this to two binomials:

(a+b)(c+d) = ?#

To do so, the distributive property must be applied twice:

$\left(a + b\right) \left(c + d\right) = \left(a + b\right) c + \left(a + b\right) d$

$= a c + b c + a d + b d$

Until the reasoning behind this is grasped, however, students may need an aid to help them remember the correct terms to multiply and add.

FOIL is a simple word to remember, and reminds the students to multiply the First terms, the Outer terms, the Inner terms, and the Last terms:

$\textcolor{red}{\text{F""irst}} : \left(\textcolor{red}{a} + b\right) \left(\textcolor{red}{c} + d\right)$
$\textcolor{red}{\text{O""uter}} : \left(\textcolor{red}{a} + b\right) \left(c + \textcolor{red}{d}\right)$
$\textcolor{red}{\text{I""nner}} : \left(a + \textcolor{red}{b}\right) \left(\textcolor{red}{c} + d\right)$
$\textcolor{red}{\text{L""ast}} : \left(a + \textcolor{red}{b}\right) \left(c + \textcolor{red}{d}\right)$

Notice that by multiplying the marked terms and adding the result, we get

$a b + a d + b c + b d$

which is equivalent to what we would get by applying the distributive property.

Note that the FOIL method does not apply to polynomials of greater size, meaning it is still important for students to learn the reasoning behind FOIL.