Hi, I’m Rob. Welcome to Math Antics.
In our last basic algebra video, we learned about polynomials.
Specifically, we learned that polynomials are chains of terms that are either added or subtracted together.
And we learned that the terms in a polynomial each have a number part and a variable part that are multiplied together.
If you don’t remember much about polynomials, you might want to re-watch the first video before you continue.
Go ahead… I’ll wait…
But even though 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, what do we like to do when things are too complicated?
Yep… 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 make the polynomial shorter.
To see how that works, have 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 ‘x to the zero’ term is also just ‘1’, so we don’t really need to show that either.
But I’m going to leave it like this for just a minute to illustrate my point.
As you can see, this polynomial has a term of every degree from zero up to four.
But, do you remember that it was okay for a polynomial to have “missing” terms?
For example, we could have a slightly different polynomial that doesn’t have a third degree term.
That makes it look like the ‘x cubed’ term gets skipped or is missing,
since the pattern goes from ‘x to the fourth’, then skips ‘x cubed’ and goes to ‘x squared’ and so on.
Well, just like 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 duplicated.
See how there are TWO terms that have an ‘x cubed’ variable part in this polynomial?
So THIS polynomial has NO ‘x cubed’ term (which is fine)
and THIS polynomial has just ONE ‘x cubed’ term (which is fine)
but THIS polynomial has TWO ‘x cubed’ terms (which is also fine)… BUT…
it’s more complicated than it needs to be!
And whenever you have terms like this…
terms that have the exact same variable part…
they can be combined into a single term.
To do that, you just add the number parts and you keep the variable part the same.
So, one ‘x cubed’ plus one ‘x cubed’ combine to form two ‘x cubed’.
What we just did there is called “Combining Like Terms”.
‘Like’ 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 a polynomial’s terms are fruit.
Yes, you heard me… fruit!
For example, have a look at this polynomial.
But let’s substitute a different kind of fruit for each different variable part.
Let’s change ‘x cubed’ to apples,
‘x squared’ to oranges,
and just plain ‘x’ to bananas.
If we do that, what would this new fruit polynomial be telling 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 these are all being added together.
So that raises 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 fruit, you can’t combine them.
[sound of machine to left of screen]
Well, unless you have a blender that is.
Ahh… but what about the middle two term?
What do we get if we add 4 oranges and 3 oranges?
That’s easy… 7 oranges!
And that means we CAN combine these two terms into a single term which makes our fruit polynomial simpler.
Now do you see 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
the way that we can if the variable parts are the same.
The mathematical reason that it works that way has to do with something called The Distributive Property,
which is the subject of a whole other video.
Alright, so if two terms in 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 get better at identifying ‘like’ terms,
let’s play a little game called “Like terms or NOT like terms?”
The first pair of terms we’ll consider is 2x and 3x.
Are they ‘like’ terms?
Yep! The variable part of both terms is the same (x) so we can combine them into a single term.
We do that by adding the number parts and keeping the variable part the same.
2 + 3 is 5, so the combined term is 5x.
Next up we have 4x and 5y.
Are these ‘like’ terms?
Well… they’re both first degree terms, but since the variables are different letters, they are NOT ‘like’ 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 ‘like’ terms and we can combine them.
Notice that one of terms is negative, so when we add the number parts we’ll end up with negative 5.
So these combine to negative five ‘x squared’.
Our next pair of terms is four ‘x squared’ and six ‘x cubed’.
Are these ‘like’ terms?
Nope! Even tough the variable is ‘x’ in both cases, the exponents are different, so the variable parts are not the same!
Next we have negative 5xy and 8yx.
Are these ‘like’ terms?
Well, at first glance, you might think that the variable parts of these terms are different
because the ‘x’ and the ‘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 re-write them so they look the same too.
There, now we can add the number parts:
negative 5 plus 8 is 3. So we wind up with the single term 3xy.
Last we have five ‘x squared y’ and five ‘y squared x’.
Now be careful with this one.
You might think that it’s like the last one where the terms are just in a different order, but look closely.
In the first term, the ‘x’ is being squared, but in the second term, the ‘y’ is being squared.
That means even if we switch the order, the exponents move with the variables so the variable parts are not the same,
which means these 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 that 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?
Yep, these two terms in the middle both have the variable ‘x’, so we can combine them.
6x minus ‘x’ would just give us 5x (since 6 - 1 is 5).
Remember, if you don’t see a number part in a term, then it's just ‘1’.
So, this polynomial started with 4 terms, but simplified to 3 terms.
‘x squared’ plus five ‘x’ plus 10.
Let’t try this one:
Sixteen
minus two ‘x cubed’
plus four ‘x’
minus ten
In this polynomial, we have a 3rd degree term, a 1st degree term and TWO constant terms.
Are those constant terms ‘like’ terms?
Absolutely! They’re both just numbers and don’t really have a variable part, so we can combine them easily.
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 terms in a polynomial as being added,
but they can have coefficients that are either positive or negative.
That’s why this negative sign stays here with the two ‘x cubed’ term… because it’s a negative term.
So this polynomial is now as simple as it can be since there are no other ‘like’ 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 to do is look to see if any of the terms are ‘like’ terms so you can combine them.
We’ll, right away you may notice that there are two constant terms in this polynomial: positive 10 and negative 4.
So let’s start by combining them into the single constant term: positive 6 (since 10 - 4 = 6)
Next, we see that there are also two 1st degree terms: negative 3x and positive ‘x’.
Those are ‘like’ terms so we can combine them: negative 3x plus 1x gives us negative 2x.
Last, we see that there are also two different terms that have the variable part ‘x squared’ so we can combine them too.
Three ‘x squared’ plus five ‘x squared’ gives us eight ‘x squared’.
So our polynomial started out 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?
Alright, so now you know how to simplify polynomials by identifying and combining ‘like’ terms.
It can sometimes be a little tricky since complicated polynomials may have many different terms
that are not necessarily in order by their degree.
That means, you may need to do some re-arranging as you look for terms that 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 off in the original polynomial so I know that I’ve already taken care of them.
Any terms that can’t be combined just come down into the simplified polynomial as is.
Oh… and to make things easier, don’t forget to treat each of the terms as either
positive or negative, depending on the sign right in front of it.
So, that’s how you simplify polynomials.
And now that you know what to do, it’s important to practice simplifying some polynomials on your own so that you really understand it.
As always… thanks for watching Math Antics, and I’ll see ya next time.
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