# Multiplying Two Binomials Using the FOIL 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:

To put this in perspective, suppose we want to multiply two arbitrary binomials, $\left( {a + b} \right)\left( {c + d} \right)$

• 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 $\left( { – \,x – 1} \right)\left( { – \,x + 1} \right)$.

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 $\left( {6x + 5} \right)\left( {5x + 3} \right)$.

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 $\left( {x – 12} \right)\left( {2x + 1} \right)$.

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 $\left( { – \,10x – 6} \right)\left( {4x – 7} \right)$.

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.

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