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• 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.

• But even though the basics of polynomials are pretty simple,

• sometimes youll 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?

• Yepwe simplify them!

• So in this video, were 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 zeroterm 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 havemissingterms?

• For example, we could have a slightly different polynomial that doesn’t have a third degree term.

• That makes it look like the ‘x cubedterm gets skipped or is missing,

• since the pattern goes from ‘x to the fourth’, then skips ‘x cubedand goes to ‘x squaredand so on.

• Well, just like there can be missing terms in a polynomial,

• there can also be EXTRA termslike in this polynomial, where the third degree term has been duplicated.

• See how there are TWO terms that have an ‘x cubedvariable part in this polynomial?

• So THIS polynomial has NO ‘x cubedterm (which is fine)

• and THIS polynomial has just ONE ‘x cubedterm (which is fine)

• but THIS polynomial has TWO ‘x cubedterms (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 cubedplus one ‘x cubedcombine to form two ‘x cubed’.

• What we just did there is calledCombining Like Terms”.

• Liketerms are terms that have exactly the same variable part.

• Butwhy 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 mefruit!

• 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 cubedto apples,

• ‘x squaredto 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 questionwhat do you get when you add 2 apples to 4 oranges?

• Wellyou 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.

• Ahhbut 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 cubedand ‘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 themliketerms and we can combine them into a single term to simplify the polynomial.

• let’s play a little game calledLike terms or NOT like terms?”

• The first pair of terms well consider is 2x and 3x.

• Are theyliketerms?

• 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 theseliketerms?

• Welltheyre both first degree terms, but since the variables are different letters, they are NOTliketerms.

• That means we can’t combine them.

• Okay, but what about these terms?

• Two ‘x squaredand negative seven ‘x squared’.

• Well the variable part in both is exactly the same. It’s ‘x squared’.

• So YES, these areliketerms and we can combine them.

• Notice that one of terms is negative, so when we add the number parts well end up with negative 5.

• So these combine to negative five ‘x squared’.

• Our next pair of terms is four ‘x squaredand six ‘x cubed’.

• Are theseliketerms?

• 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 theseliketerms?

• 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 youve had some practice identifyingliketerms,

• let’s look at some complicated polynomials that we can simplify by combining anyliketerms 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 squaredplus 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 termsliketerms?

• Absolutely! Theyre 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 cubedtermbecause it’s a negative term.

• So this polynomial is now as simple as it can be since there are no otherliketerms.

• 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 areliketerms so you can combine them.

• Well, 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 areliketerms 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 squaredso we can combine them too.

• Three ‘x squaredplus five ‘x squaredgives 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 squaredminus 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 combiningliketerms.

• 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.

• Ohand 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 alwaysthanks for watching Math Antics, and I’ll see ya next time.