Multiplication with the Distributive Property
Polynomial multiplication is based on a principle called the Distributive Property. If you look below at how it's defined, it looks deceptively simple. Be sure to pay close attention to the examples of how negative signs work when applying this property - students often run into trouble with that situation.
a(b + c) = ab + ac
You can get rid of parentheses in an expression by taking the expression that's outside the parentheses and multiplying everything inside the parentheses by it.
I think the best way to approach this material is by looking at some specific examples.
Quick Definition - "Simplify"
"Simplify" is one of the vaguest terms you'll see in mathematics. What constitutes a 'simpler' version of an expression can sometimes be a matter of opinion. For example 3(x + 1) and 3x + 3 are both equal to each other but I'd be hard pressed to give you an objective reason why one was 'simpler' than the other. There are a few universal rules that you should remember:
- Fractions should be completely reduced, e.g. 1/2 is simpler than 2/4.
- Exponents should be combined whenever possible, e.g. x5 is simpler than x2x3.
- Negative exponents should always be converted to positive ones, e.g. 1 / x is simpler than $ x^{-1}$.
- Radicals should be reduced as much as possible, e.g. $4\sqrt{3}$ is simpler than $\sqrt{48}$
- Denominators should be rationalized, e.g. $\sqrt{3} / 3$ is simpler than $1 / \sqrt{3}$.
Example 1
Simplify 2(4x + 3).
The Distributive Property tells me that I can simplify this by taking the 2 outside the parentheses and "distributing" it to both of the terms inside the parentheses.
2(4x + 3) = 2·4x + 2·3 = 8x + 6
Example 2
Simplify -2(x3 - 3x2).
This one is a little trickier because of the negative signs. I'm going to distribute the -2 just like I distributed the +2 in the previous example.
-2(x3 - 3x2) = -2·x3 - (-2)·3x2
Notice that I didn't do anything with the minus sign inside the parentheses. At this point, all I'm trying to do is distribute the -2. We'll worry about the minus in a later step. Now I'm going to multiply out all the numbers. In this case, there's only one pair - the -2 and the 3 on the last term.
-2(x3 - 3x2) = -2x3 - (-6x2)
Now, I'm going to straighten out the minus sign. Remember that a negative times a negative is a positive so anywhere I see two negatives, I'm going to change them to a plus.
-2(x3 - 3x2) = -2x3 - (-6x2) = -2x3 + 6x2
Example 3
Simplify 2x2(-3x3 + 3y + 1).
Don't be intimidated by there being a variable outside the parentheses. You distribute the entire 2x2 just like the numbers in the previous examples.
| (2x2)(-3x3) + (2x2)(3y) + 1(2x2) | I took the 2x2 and multiplied every term inside the parentheses by it. |
| 2·(-3)·x2·x3 + 2·3x2y + 1·2x2 | I rearranged the parts of each term so that the "like" parts where together. For example in the first term, I moved the numbers together and the variable parts together. This is just to make it clear which parts I'm going to multiply together on the next step. (You could skip this and go directly to the next step if it's clear to you which parts can be multiplied together.) |
| -6x5 + 6x2y + 2x2 | I multiplied the numbers together and multiplied the variable parts by adding their exponents. |
