Multiplying Binomial Expressions

Motivation

Assume that a family is considering extending the size of their house and that its current width is 12 meters and its current length is 10 meters.

The family is not yet sure how much they wish to extend the house so they assume its width is to be extended x meters and that its length is to be extended y meters.

The new width of the house will be and the new length of the house will be . The new area of the house will therefore be given by the formula or .

To deal with such situations, we need to be able to multiply expressions such as .

Product of Sums

Solution: This can be best visualized by the following rectangle.

There are two approaches to this problem:

• The first is to obtain the product can be obtained by finding the area of the large rectangle.

Using this technique, the associated area would be .

• The second is to find the area of all of the internal rectangles and add them together.

In this case, the area is equal to . As the area of the same square cannot change, the two approaches will give the same answer. And in terms of multiplying binomial expressions, we can conclude that .

Example #2: Find the product of (x + 3) (x + 2).

Solution: This can be best visualized by the following rectangle.

There are two approaches to this problem:

• The first is to obtain the product can be obtained by finding the area of the large rectangle.

Using this technique, the formula for the associated area is and cannot be simplified.

• The second is to find the area of all of the internal rectangles and add them together.

In this case, the formula for the associated area is . As the area of the same square cannot change, the two approaches will give the same answer. Hence, we can conclude that .

Solution: This can be best visualized by the following rectangle.

There are two approaches to this problem:

• The first is to obtain the product can be obtained by finding the area of the large rectangle.

Using this technique, the formula for the associated area is and cannot be simplified.

• The second is to find the area of all of the internal rectangles and add them together.

In this case, the formula for the associated area is . As the area of the same square cannot change, the two approaches will give the same answer. Hence, we can conclude that .

Products of differences

Example: (x - 2) (x - 3)

However, a simpler method is to pretend negatives can be added like other numbers.

Example #1: Find the product (x - 2) (x - 3)

Solution: We first rewrite the difference as (x + (-2))(x + (-3)).

There are two approaches to this problem:

• The first is to obtain the product can be obtained by finding the area of the large rectangle.

Using this technique, the formula for the associated area is and cannot be simplified.

• The second is to find the area of all of the internal rectangles and add them together.

In this case, the formula for the associated area is . As the area of the same square cannot change, the two approaches will give the same answer. Hence, we can conclude that .

Example #2: Find the product (2x - 1) (3x - 3)

• The first is to obtain the product can be obtained by finding the area of the large rectangle.

• The second is to find the area of all of the internal rectangles and add them together.