Algebra Basics: Simplifying Polynomials - Math Antics

Hi, I'm Rob. Welcome to Mathematical Mischief. In our last basic

algebra

video, we learned about

polynomials

. Specifically, we learned that

polynomials

are strings of terms that are added or subtracted. And we learned that each of the terms of a polynomial has a numerical part and a variable part that multiply together. If you don't remember much about polynomials, you may want to rewatch the first video before continuing. Go ahead... I'll wait... But while the

basics

of polynomials are pretty simple, sometimes you'll come across polynomials that are more complicated than they really need to be. And in

math

ematics, what do we like to do when things are too complicated?

Yes… we simplify them! So in this video, we're going to learn how to simplify polynomials. Simplifying a polynomial involves identifying terms that are similar enough that they can be combined into a single term to shorten the polynomial. To see how it works, take a look at this basic polynomial that follows an easy-to-recognize pattern. Of course, as I mentioned in the last video, we don't really need to show the coefficients of each term if they are just "1" like we have here. And the term "x to the power of zero" is also simply "1", so we don't need to prove it either.

But I'm going to leave it at that just for a minute to illustrate my point. As you can see, this polynomial has one term of each degree from zero to four. But remember how it was okay for a polynomial to have missing terms? For example, we could have a slightly different polynomial that does not have a third degree term. That makes it look like the term "x cubed" is skipped or missing, as the pattern goes from "x to the quarter", then skips "x cubed" and goes to "x squared" and so on. Well, just as there can be missing terms in a polynomial, there can also be EXTRA terms... like in this polynomial, where the third degree term has been doubled.

See how there are TWO terms that have a variable part 'x cubed' in this polynomial? So THIS polynomial has NO 'x cubed' term (which is fine) and THIS polynomial has only ONE 'x cubed' term (which is fine) but THIS polynomial has TWO 'x cubed' terms (which that's fine too)... BUT... It's more complicated than necessary! And as long as you have terms like this... terms that have exactly the same variable part... they can be combined into a single term. To do that, simply add the numerical parts and keep the variable part the same. So, one “x cubed” plus one “x cubed” combine to make two “x cubed.” What we just did there is called "Combine Like Terms." "Similar" terms are terms that have exactly the same variable part.

But… why can we combine them? Well, to understand that, I like to pretend that the variable parts of the terms of a polynomial are fruits. Yes, you heard me… fruit! For example, take a look at this polynomial. But let's replace each different variable part with a different type of fruit. Let's change 'x cubed' to apples, 'x squared' to oranges and simply 'x' to bananas. If we do that, what would this new fruit polynomial tell us? Well, this first term represents 2 apples, the next term is 4 oranges, the next term is 3 oranges, and the last term is 5 bananas.

And all of these are adding up. So that begs the question…what do you get when you add 2 apples to 4 oranges? Well… you get… 2 apples and 4 oranges! Since they are different fruits, you cannot combine them. Well, unless you have a blender. Ahh... but what about the second term in the middle? What do we get if we add 4 oranges and 3 oranges? It's that easy... 7 oranges! And that means we CAN combine these two terms into one, which simplifies our fruit polynomial. Do you now understand why the variable parts of a term have to be exactly the same in order to combine them?

If the variable parts are different (like "x cubed" and "x squared"), then they represent different things, so we can't group them into a single term in the same way we would if the variable parts are the same. The

math

ematical reason it works that way has to do with something called The Distributive Property, which is the topic of a completely different video. Okay, so if two terms of a polynomial have exactly the same variable part, then we call them "like" terms and we can combine them into a single term to simplify the polynomial. And to help you better identify "similar" terms, let's play a little game called "Similar Terms or NOT Similar Terms?".

The first pair of terms we will consider is 2x and 3x. Are they “similar” terms? Yeah! The variable part of both terms is the same (x) so we can combine them into a single term. We do this by adding the numerical parts and keeping the variable part the same. 2 + 3 is 5, so the combined term is 5x. Next we have 4x and 5y. Are these terms “similar”? Well... they are both first degree terms, but since the variables are different letters, they are NOT "similar" terms. That means we can't combine them. Okay, but what about these terms? Two 'x squared' and negative seven 'x squared'.

Well, the variable part in both is exactly the same. It's "x squared." So YES, these are similar terms and we can combine them. Notice that one of the terms is negative, so when we add the numerical parts we will end up with negative 5. These then combine to get negative five 'x squared'. Our next pair of terms is four "x squared" and six "x cubed." Are these terms “similar”? No! Although the variable is "x" in both cases, the exponents are different, so the parts of the variable are not equal. Next we have negative 5xy and 8yx. Are these terms “similar”?

Well, at first glance, you might think that the variable parts of these terms are different because the 'x' and 'y' are in a different order. But remember, multiplication has the commutative property, so the order doesn't matter. xy is the same as yx, so we can rewrite them to look the same as well. There, we can now add the numerical parts: negative 5 plus 8 is 3. So we end up with the single term 3xy. Finally we have five 'x squared y' and five 'y squared x'. Now be careful with this one. You might think it's like the last one, where the terms are in a different order, but look closely.

In the first term, the "x" is squared, but in the second term, the "y" is squared. That means that even if we change the order, the exponents move with the variables, so the parts of the variables are not the same, which means they are NOT like terms. Alright, now that you've had some practice identifying "like" terms, let's look at some complicated polynomials that we can simplify by combining any "like" terms we find. Here's our first example: 'x squared' plus six 'x' minus 'x' plus ten Do you see any terms that have the same variable part? Yes, these two middle terms have the variable "x", so we can combine them. 6x minus 'x' would simply give us 5x (since 6 - 1 is 5).

Remember, if you don't see a numerical part in a term, then it's just "1." So this polynomial started with 4 terms, but was simplified to 3 terms. 'x squared' plus five 'x' plus 10. Let's try this: Sixteen minus two 'x cubed' plus four 'x' minus ten In this polynomial, we have one third-degree term, one first-degree term, and TWO constant terms. Are those constant terms “similar” terms? Absolutely! They are both just numbers and don't actually have a variable part, so we can easily combine them. This term is positive 16 and this term is negative 10, so if you add them together, you end up with positive 6.

Remember, it's best to think of all the terms in a polynomial as adding, but they can have coefficients that are positive or negative. That is why this negative sign remains here with the two “x cubed” term… because it is a negative term. So this polynomial is now as simple as can be, since there are no other "similar" terms. Ready for an even more complicated polynomial? Three 'x squared' plus ten minus three 'x' plus five 'x squared' minus four plus 'x' This polynomial has SIX terms, and when you get a long polynomial like this, the first thing you do is look to see if any of the terms are "similar" terms, so you can combine them.

Well, right away you can notice that there are two constant terms in this polynomial: positive 10 and negative 4. So let's start by combining them into a single constant term: positive 6 (since 10 - 4 = 6). Note that there are also two first degree terms: negative 3x and positive 'x'. Those are similar terms, so we can combine them: negative 3x plus 1x gives us negative 2x. Lastly, we see that there are also two different terms that have the variable part 'x squared' so we can also combine them. Three 'x squared' plus five 'x squared' gives us eight 'x squared'. So our polynomial started with six terms, but we were able to simplify it to just three terms: eight 'x squared' minus two 'x' plus six.

That almost made

algebra

seem fun, didn't it? Okay, now you know how to simplify polynomials by identifying and combining "like" terms. It can be a bit complicated at times, as complicated polynomials can have many different terms that are not necessarily ordered according to their degree. That means you may need to rearrange some terms as you search for terms you can combine. I like to look for pairs that I can combine and then once I combine them into a single term in my simplified polynomial, I cross them out in the original polynomial so I know I've already taken care of them.

Any terms that cannot be combined simply reduce to the simplified polynomial as is. Oh... and to make things easier, don't forget to treat each of the terms as positive or negative, depending on the sign in front of it. This is how polynomials are simplified. And now that you know what to do, it's important that you practice

simplifying

some polynomials on your own so that you really understand it. As always...thanks for watching Math Antics and I'll see you next time. More information at www.math

antics

.com