Back To CourseELM: CSU Math Study Guide
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Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
Negative signs - they have incredible power. They can reduce a number to not just nothing, but less than nothing. It's one thing if you had 3 oranges and now you have 0. What if you had -3 oranges? That's like having 3 black holes where your oranges were. It's weird. It's like negative signs create Bizarro World numbers of regular numbers.
That negative sign is thrown on them like the goatee on evil Spock. You know you can't trust someone with a goatee. Well, except me. Or, am I an evil version of myself? Oh man. Anyway, negative signs. They can also wreak havoc in algebraic expressions. They can pop up anywhere and take an otherwise straightforward problem and make it confusing. But, fear not. We can handle a few negative signs in some algebraic expressions like the algebra superheroes we are. Let's learn how.
First, we should familiarize ourselves with simplifying algebraic expressions, or making expressions simpler by using the distributive property and combining like terms.
The distributive property, of course, is one of our algebraic expression superpowers. It's when we take a(b + c) and make it equal to (ab) + (ac). This law keeps multiple variables inside parentheses from slowing us down. If you see something like 3(2m + 5n), use the distributive property and distribute that 3 across the parentheses, leaving you with 6m + 15n.
Then there's combining like terms. This is when we join terms with the same variable. You know how at the end of superhero movies the villain is defeated, but the city is also kind of destroyed? It takes a special kind of superhero to put those buildings back together. You need to know where everything goes and match up the right pieces.
When you have 2x + x^2 + 6x + 3x^2, we match up 2x and 6x to get 8x. Then, we match x^2 and 3x^2 to get 4x^2. That makes our rebuilt city, and our simplified expression, 8x + 4x^2.
Okay, we know what we need to do. Let's go handle some negative signs. Let's start with some distributive property ones.
Here's one: -(3s + 2t). Can we simplify this? We can't add 3s to 2t. But, we can distribute the number outside the parentheses. 'Wait,' you might say. 'There is no number there. There's just that negative sign.'
Remember, that negative sign is really a -1 with some cloaking powers, and we can totally distribute it. -1 * 3s is -3s. And, -1 * 2t is -2t. We put that together, and we get -3s + -2t, which is -3s - 2t.
Here's another: -(5p - 2r). Okay, this looks just like the last one, but with one important difference. Yep, that minus sign. I think the easiest way to handle this is to treat the 5p, 2r and minus sign as unique parts.
We multiply -1 * 5p and get -5p. Then, we multiply -1 * 2r and get -2r. We put those back in our expression and we have -5p - (-2r). What's - (-2r)? + 2r. So, our final expression is -5p + 2r. You could also think of the original expression as -1(5p + (-2r)). Either way, don't lose track of that minus sign!
Okay, how about one practice with combining like terms? Here's one: 3y - 4y. Here we have two terms, 3y and 4y. And, they just happen to be like terms - awesome. What do we get when we take 4 from 3? -1. So, what's 3y - 4y? -y. That's it.
That wasn't really superhero-level, was it? How about this: 3p - 9p^2 - 6p^2 + 4 - 2p + p^2. We want to get the like terms next to each other by shuffling things around. But, when we have a mix of plus and minus signs, we need to be very careful that we don't lose any.
What like terms do we have? 3p and 2p. So, let's move the 2p over. But wait, it's a -2p. That's better (shown below). We also have this -9p^2, -6p^2 and +p^2. Let's move the +p^2 over, as shown below. We still have that 4, but we can't do anything with that.
Now, let's combine the like terms. 3p - 2p is just p. -9p^2 - 6p^2 is -15p^2. A common mistake is to just see the 9p^2 and do 9 - 6 to get +3p^2. If you do that, you're letting a negative sign get away. Then you're just going to have to deal with it in the sequel. And, nobody wants to see the same villain twice.
So, we have -15p^2 + p^2. What's -15 + 1? -14. So, our simplified expression is p - (don't forget that minus) 14p^2 + 4.
Now we've practiced both the distributive property and combining like terms. Let's put them together for a final, epic battle.
[-2(3x^2 - 5xy) - 3x(x + 2y)] - [-x(4x + y) - y(3 - 2x)]
Whoa. That's a monster. Let's first see if there are any like terms inside parentheses that we can combine. Not here, or here, or here, or here (please see image below). Okay, it's time to bring out the distributive property. Now, there are a lot of minus signs. Let's take it slow and not miss any.
First, we distribute the -2:
-2 * 3x^2 is -6x^2
-2 * -5xy is +10xy
Now the 3x:
3x * x is 3x^2
3x * 2y is 6xy
Don't forget the negative sign, which makes it -3x^2 - 6xy.
So, the first half of our expression is -6x^2 + 10xy - 3x^2 - 6xy.
Let's look at the second half:
-x * 4x is -4x^2
-x * y is -xy
y * 3 is 3y
y * -2x is -2xy
So, we have -4x^2 - xy and 3y - 2xy. But, don't forget this sneaky minus sign here (see below).
So, it's -4x^2 - xy - 3y + 2xy.
Before we put these two halves together, remember that they're joined by a minus sign. So, we also need to distribute that across the second half. That will give us +4x^2 + xy + 3y - 2xy.
So, now we have, wait for it, -6x^2 + 10xy - 3x^2 - 6xy + 4x^2 + xy + 3y - 2xy. Our monster is apparently a shape-shifter. Well, we've been conquering it to get here. Let's put our like terms together and finish it.
Let's move the -3x^2 and the +4x^2 over with the -6x^2. When we do that, we have the +10xy next to the -6xy and +xy. Let's drag the -2xy over (please see below for these transitions).
And, now it's time to combine. -6x^2 - 3x^2 is -9x^2. Add 4x^2, and we have -5x^2. Now, 10xy - 6xy is 4xy. If we add xy and subtract 2xy, we have +3xy. That makes our simplified expression -5x^2 + 3xy + 3y. Remember what we started with? Yeah, I think we won that battle.
In summary, negative signs can wreak havoc on complicated algebraic expressions. The principles of what we're doing when we simplify, though, don't change. Remember, simplifying algebraic expressions is making expressions simpler by utilizing both the distributive property and combining like terms. The distributive property can be boiled down to a(b + c) = (ab) + (ac). And, combining like terms is joining terms with the same variable. When our expressions have negative signs, watch them closely. As long as you never lose sight of them, there's no expression you can't simplify.
Once you've completed this lesson, you'll be able to properly use the distributive property. You could know how to combine like terms to simplify complicated expressions that contain negative signs.
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Back To CourseELM: CSU Math Study Guide
16 chapters | 140 lessons