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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.

  • Go ahead… I’ll wait

  • 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.

  • And to help you get better at identifyingliketerms,

  • 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.

  • Learn more at www.mathantics.com

Hi, I’m Rob. Welcome to Math Antics.

Subtitles and vocabulary

A2 BEG US polynomial term variable squared combine cubed

Algebra Basics: Simplifying Polynomials - Math Antics

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    Yassion Liu   posted on 2016/07/22
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