Simplify two 𝑥 cubed minus four 𝑥 squared minus nine minus five 𝑥 cubed minus three 𝑥 plus one.
The first thing we can do is go ahead and copy the problem down exactly as you see it here. In order to simplify this problem, we’re going to have to deal with the parentheses; we’re going to have to deal with this grouping. The first way we’re going to get the parentheses out of the way is by using the distributive property. And you might be thinking, “wait! there is nothing to distribute here.” But do you know what we want to distribute? We want to distribute this negative across each term or more specifically we wanna distribute a negative one across five 𝑥 cubed, negative three 𝑥, and one.
Here’s what we’ve done. We’ve changed the subtraction sign that we started with to addition and then we’re multiplying negative one by what’s in the parentheses. First step, copy down everything in the first parentheses exactly how it’s written. Now, we need to multiply negative one by five 𝑥 cubed, which gives us negative five 𝑥 cubed. We multiply negative one times negative three 𝑥, and we get positive three 𝑥, multiplying negative one by one, which is negative one.
If you look closely what we’ve done here is we’ve done two things: we’ve got rid of the parentheses and we’ve also changed the sign on each of the terms. So we changed five 𝑥 cubed to negative five 𝑥, changed three- negative three 𝑥 to positive three 𝑥, and we changed positive one to negative one.
Again, we have another set of parentheses that we need to deal with, but this time we won’t use the distributive property. Here, we’ll need the associative property of addition. And that tells us that we’re only adding; the order does not matter. Now maybe, I’ve really lost you because I’m talking about the associative property of addition and you see some subtraction in this problem. We’re going to fix that. Two 𝑥 cubed doesn’t have a negative, doesn’t have subtraction, we can just bring that down. Then, we’ll write plus negative four 𝑥 squared plus negative nine. We can copy down our negative five 𝑥 cubed and our three 𝑥. Then, we’ll add plus negative one.
Now, we’ll really use our associative property that says we can add these values in any order; it also means that we can regroup them. And to simplify, we’ll need to regroup based on like terms. What I mean by that is we’ll need to add two 𝑥 cubed to negative five 𝑥 cubed. They are alike because their variable is taken to the same power. We can only add like terms, so we can add two 𝑥 cubed plus negative five 𝑥 cubed.
Now, I’m looking at the negative four 𝑥 squared. There are no more 𝑥 squared terms, which means there’s nothing I can add to negative four 𝑥 squared. So we’ll bring it down exactly as it is. Is there any other alike term? Is there any term that’s like three 𝑥? There’s not. So we bring down the three 𝑥 and we can’t simplify that term any further. Finally, we’re left with whole numbers. We can add negative nine and negative one together.
Our final step here will be to go ahead and add each term that can be added together. Two 𝑥 cubed plus negative five 𝑥 cubed equals negative three 𝑥 cubed. Negative four 𝑥 squared is already simplified; you just bring it down. The same thing with three 𝑥, it’s simplified; bring it down. Negative nine plus negative one equals negative 10, so we can write that as minus 10. By using properties of addition, the distributive and the associative property, we simplified this polynomial into negative three 𝑥 cubed minus four 𝑥 squared plus three 𝑥 minus 10.