FOIL Method

The term FOIL is a widely recognized acronym in mathematics. Each letter in FOIL represents a specific word, and here’s how it breaks down.

F = first

O = outer

I = inner

L = last

The words first, outer, inner, and last all pertain to the terms in a binomial. The best way to demonstrate this concept is to go over a few examples.

Example 1: Multiply the binomials using the FOIL method.

$$\left( {x + 3} \right)\left( {x + 4} \right)$$

If we look at the binomials above, we see that the first term in each binomial is \(x\). The outer terms are \(x\) and \(4\), while the inner terms are \(3\) and \(x\). Lastly, the last terms are \(3\) and \(4\). We simply need to multiply these terms and then combine any like terms.

$$F \to \left( x \right)\left( x \right) = {x^2}$$

$$O \to \left( x \right)\left( 4 \right) = 4x$$

$$I \to \left( 3 \right)\left( x \right) = 3x$$

$$L \to \left( 3 \right)\left( 4 \right) = 12$$

Now, we write all terms horizontally.

$${x^2} + 4x + 3x + 12$$

Finally, combine the like terms which are \(4x\) and \(3x\).

$${x^2} + 7x + 12$$

Example 2: Multiply the binomials using the FOIL method.

$$\left( {y + 2} \right)\left( {y – 6} \right)$$

Let’s do an example where one of the binomial has a minus sign in the middle.

$$F \to \left( y \right)\left( y \right) = {y^2}$$

$$O \to \left( y \right)\left( { – 6} \right) =  – 6y$$

$$I \to \left( 2 \right)\left( y \right) = 2y$$

$$L \to \left( 2 \right)\left( { – 6} \right) =  – 12$$ 

Putting them together, we have

$${y^2} – 6y + 2y – 12$$

Then combing like terms, we get

$${y^2} – 4y – 12$$

Example 3: Multiply the binomials using the FOIL method.

$$\left( {k – 1} \right)\left( {k – 2} \right)$$

Let’s try an example where both of the binomials have a negative sign in the middle.

Multiplying the first (F) terms:

$$\left( k \right)\left( k \right) = {k^2}$$

Multiplying the outer (O) terms:

$$\left( k \right)\left( { – 2} \right) =  – 2k$$

Multiplying the inner (I) terms:

$$\left( { – 1} \right)\left( k \right) =  – k$$

Multiplying the last (L)  terms:

$$\left( { – 1} \right)\left( { – 2} \right) = 2$$

Putting all the products together, we obtain

$${k^2} – 2k – k + 2$$

Finally, combine the similar terms \(-2k\) and \(-k\). We know that \( – 2k – k =  – 3k\).

$${k^2} – 3k + 2$$