Negative Signs And Simplifying Algebraic Expression

Negative Signs

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.

Simplifying Algebraic Expressions

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.

Distributive Property Practice

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!