## Multiplying a Binomial By a Binomial

Contents

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a **binomial** times a binomial. We will start by using the Distributive Property.

## Multiply a Binomial by a Binomial Using the Distributive Property

Look at this example from the previous lesson, where we multiplied a binomial by a **monomial**.

We distributed the p to get: | |

What if we have (x + 7) instead of p? | |

Distribute (x + 7). | |

Distribute again. | |

Combine like terms. |

Notice that before combining like terms, you had four terms. You multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.

## Example

Multiply: \(\left(y+5\right)\left(y+8\right).\)

### Solution

Distribute (y + 8). | |

Distribute again | |

Combine like terms. |

## Example

Multiply: \(\left(2y+5\right)\left(3y+4\right).\)

### Solution

Distribute (3y + 4). | |

Distribute again | |

Combine like terms. |

## Example

Multiply: \(\left(4y+3\right)\left(2y-5\right).\)

### Solution

Distribute. | |

Distribute again. | |

Combine like terms. |

## Example

Multiply: \(\left(x+2\right)\left(x-y\right).\)

### Solution

Distribute. | |

Distribute again. | |

There are no like terms to combine. |

## Multiply a Binomial by a Binomial Using the FOIL Method

Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a **trinomial**, but sometimes, like in the example above, there are no like terms to combine.

Let’s look at the last example again and pay particular attention to how we got the four terms.

\(\begin{array}{c}\hfill \left(x-2\right)\left(x-y\right)\hfill \\ \hfill {x}^{2}-xy-2x+2y\hfill \end{array}\)

Where did the first term, \({x}^{2}\), come from?

We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘**F**irst, **O**uter, **I**nner, **L**ast’. The word FOIL is easy to remember and ensures we find all four products.

\(\begin{array}{c}\hfill \left(x-2\right)\left(x-y\right)\hfill \\ \hfill \underset{F\phantom{\rule{2em}{0ex}}O\phantom{\rule{2.5em}{0ex}}I\phantom{\rule{2.5em}{0ex}}L}{{x}^{2}-xy-2x+2y}\hfill \end{array}\)

Let’s look at \(\left(x+3\right)\left(x+7\right)\).

Distibutive Property | FOIL |

Notice how the terms in third line fit the FOIL pattern.

Now we will do an example where we use the FOIL pattern to multiply two binomials.

### Example: How to Multiply a Binomial by a Binomial using the FOIL Method

Multiply using the FOIL method: \(\left(x+5\right)\left(x+9\right).\)

### Solution

We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!

### Multiply two binomials using the FOIL method

When you multiply by the FOIL method, drawing the lines will help your brain focus on the pattern and make it easier to apply.

## Example

Multiply: \(\left(y-7\right)\left(y+4\right).\)

### Solution

## Example

Multiply: \(\left(4x+3\right)\left(2x-5\right).\)

### Solution

The final products in the last four examples were trinomials because we could combine the two middle terms. This is not always the case.

## Example

Multiply: \(\left(3x-y\right)\left(2x-5\right).\)

### Solution

Multiply the First. | |

Multiply the Outer. | |

Multiply the Inner. | |

Multiply the Last. | |

Combine like terms—there are none. |

Be careful of the exponents in the next example.

## Example

Multiply: \(\left({n}^{2}+4\right)\left(n-1\right).\)

### Solution

Multiply the First. | |

Multiply the Outer. | |

Multiply the Inner. | |

Multiply the Last. | |

Combine like terms—there are none. |

## Example

Multiply: \(\left(3pq+5\right)\left(6pq-11\right).\)

### Solution

Multiply the First. | ||

Multiply the Outer. | ||

Multiply the Inner. | ||

Multiply the Last. | ||

Combine like terms—there are none. |

## Multiply a Binomial by a Binomial Using the Vertical Method

The FOIL method is usually the quickest method for multiplying two binomials, but it *only* works for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.

Now we’ll apply this same method to multiply two binomials.

## Example

Multiply using the Vertical Method: \(\left(3y-1\right)\left(2y-6\right).\)

### Solution

It does not matter which binomial goes on the top.

Notice the partial products are the same as the terms in the FOIL method.

We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The methods are listed here all together, to help you remember them.

### Multiplying Two Binomials

To multiply binomials, use the:

- Distributive Property
- FOIL Method
- Vertical Method

Remember, FOIL only works when multiplying two binomials.