# FOIL (First, Outer, Inner and Last) Method

FOIL (the acronym for first, outer, inner and last) method is an efficient way of remembering how to multiply two binomials in a very organized manner.

The word FOIL is an acronym that stands for

** **

- The
**first**means that we multiply the terms which occur in the first position of each binomial.

- The
**outer**means that we multiply the terms which are located in both ends (outermost) of the two binomials when written side-by-side.

- The
**inner**means that we multiply the middle two terms of the binomials when written side-by-side.

- The
**last**means that we multiply the terms which occur in the last position of each binomial.

- After obtaining the four (4) partial products coming from the
**first**,**outer**,**inner**and**last**, we simply add them together to get the final answer.

## Examples of How to Multiply Binomials using the FOIL Method

**Example 1**: Multiply the binomials \left( {x + 5} \right)\left( {x - 3} \right) using the FOIL Method.

- Multiply the pair of terms coming from the
**first**position of each binomial.

- Multiply the
**outer**terms when the two binomials are written side-by-side.

- Multiply the
**inner**terms when the two binomials are written side-by-side.

- Multiply the pair of terms coming from the
**last**position of each binomial.

- Finally, simplify by combining like terms. I see that we can combine the two middle terms with variable x.

**Example 2**: Multiply the binomials \left( {3x - 7} \right)\left( {2x + 1} \right) using the FOIL Method.

If the first presentation on how to multiply binomials using FOIL doesn’t make sense yet. Let me show a different way. The idea is to expose you to different ways on how to address the same type of problem with a different approach.

- Multiply the
**first**terms

- Multiply the
**outer**terms

- Multiply the
**inner**terms

- Multiply the
**last**terms

After applying the FOIL, we arrive at this polynomial which we can simplify by combining similar terms. The two middle x-terms can be subtracted to get a single value.

**Example 3**: Multiply the binomials \left( { - \,4x + 5} \right)\left( {x + 1} \right) using the FOIL Method.

Another way of doing this is to list the four partial products, and then add them together to get the answer.

- Multiply the
**first**terms

- Multiply the
**outer**terms

- Multiply the
**inner**terms

- Multiply the
**last**terms

Get the sum of the partial products, and then combine similar terms.

**Example 4**: Multiply the binomials \left( { - \,7x - 3} \right)\left( { - \,2x + 8} \right) using the FOIL Method.

**Solution**:

- Multiply the
**first**terms

- Multiply the
**outer**terms

- Multiply the
**inner**terms

- Multiply the
**last**terms

Finally, combine like terms to finish this off!

**Example 5**: Multiply the binomials \left( { - \,x - 1} \right)\left( { - \,x + 1} \right) using the FOIL Method.

**Solution**:

- Multiply the
**first**terms

- Multiply the
**outer**terms

- Multiply the
**inner**terms

- Multiply the
**last**terms

Notice that the middle two terms cancel each other out!

**Example 6**: Multiply the binomials \left( {6x + 5} \right)\left( {5x + 3} \right) using the FOIL Method.

**Solution**:

- Product of the
**first**terms

- Product of the
**outer**terms

- Product of the
**inner**terms

- Product of the
**last**terms

Add the two middle x-terms, and we are done!

**Example 7**: Multiply the binomials \left( {x - 12} \right)\left( {2x + 1} \right) using the FOIL Method.

**Solution**:

- Product of the
**first**terms

- Product of the
**outer**terms

- Product of the
**inner**terms

- Product of the
**last**terms

After expanding the binomials, combine like terms to get the final answer!

**Example 8**: Multiply the binomials \left( { - \,10x - 6} \right)\left( {4x - 7} \right) using the FOIL Method.

**Solution**:

- Multiply the
**first**terms

- Multiply the
**outer**terms

- Multiply the
**inner**terms

- Multiply the
**last**terms

After distributing the terms of the two binomials using the FOIL method, combine like terms to get the final answer.

**You might also be interested in:**

Adding and Subtracting Polynomials

Dividing Polynomials using Long Division Method

Dividing Polynomials using Synthetic Division Method

Multiplying Polynomials