5B Multiplying Terms with Associative and Distributive Properties
5B Multiplying Terms with Associative and Distributive Properties
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Remember a term is a product of numbers and/or letters. A term can be a letter by itself or a number by itself or a product of numbers and letters. The key thing to remember about terms is that we can combine terms by addition or subtraction only when they are “like” terms, that is ones that have the exact same letter part.
Many formulas have unlike terms that cannot be combined into a single term. For example the perimeter of a rectangle is L+W+L+W which when simplified gives 2L+2W. Suppose we want to triple the length and width of a rectangle. This would triple the perimeter 3(2L+2W). How would we simplify this formula? This is what we will try to figure out in this section.
Let’s start by thinking about multiplying terms. For example, look at 4(5y). The associative property says that a(bc)=(ab)c. This means that . Notice we cannot combine unlike terms into a single term, but we can always multiply terms since they are a product. Multiply the numerical coefficients and the variable parts together. Look at . This is the same as or . Notice also that when we multiply two terms, our answer is just one term.
Multiply the following terms with your instructor.
Let’s look at the example of tripling the perimeter of a rectangle: 3(2L + 2W). The problem again is that 2L and 2W are not like terms so they cannot be combined into a single term. So how do we multiply by 3? The answer is by using the distributive property. The distributive property is a(b+c) = ab+ac. So to multiply 3(2L + 2W), we multiply 3 times 2L and 3 times 2W. Hence . So the formula for triple the perimeter is 6L+6W.
Let’s try another example. . It is often helpful to rewrite subtraction as adding a negative. In this example . Now we use the distributive property to multiply. . So the simplified formula is .
Sometimes we want to subtract a parenthesis. For example, look at . The key is to subtract all the terms inside the parenthesis. This is often called distributing the negative. You can also think of it like multiplying by -1. When we distribute the negative we get the following:
. Notice that when we distribute the negative all the signs of the terms inside the parenthesis have changed to the opposite. -3x became +3x and +8 became -8.
Let’s look at a more complicated example of simplifying. . In problems like this, order of operations comes into play. In order of operations, we simplify parenthesis first. 9y-4 and -2y+8 are both as simplified as possible. They are not like terms so we cannot combine them into a single term. Next we do the multiplications. It is very helpful to rewrite subtractions as adding the negative. So . Now we use the distributive property to multiply.
We are not quite finished since now there are some like terms we can add. Remember the y terms with the y terms and the constants with the constants.
So when simplified completely this formula is -44y-16. Remember adding -16 is the same as subtracting 16.
Simplify the following with your instructor by using the distributive property. Remember to simplify completely.
Practice Problems Section 5B
Simplify the following using the associative property. Be sure to simplify completely.
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Simplify the following using the distributive property. Be sure to simplify completely.
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Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.