Algebra Basics: Simplifying Polynomials - Math Antics - Free Educational videos for Students in K-12

Algebra Basics: Simplifying Polynomials - Math Antics - By Mathantics

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00:03	Uh huh . Hi , I'm rob . Welcome to
00:07	Math antics And our last basic algebra video , we
00:10	learned about polynomial specifically , we learned that polynomial are
00:14	chains of terms that are either added or subtracted together
00:18	. And we learned that the terms in a polynomial
00:21	each have a number part and a variable part that
00:23	are multiplied together . If you don't remember much about
00:26	polynomial , you might want to rewatch the first video
00:29	before you continue . Go ahead . Wait . Even
00:43	though the basics of polynomial are pretty simple , sometimes
00:46	you'll come across polynomial that are more complicated than they
00:49	really need to be . And in math , what
00:51	do we like to do when things are too complicated
00:54	, yep We simplify them . So in this video
00:57	we're going to learn how to simplify polynomial . Simplifying
01:00	a polynomial involves identifying terms that are similar enough that
01:04	they can be combined into a single term to make
01:07	the polynomial shorter to see how that works . Have
01:10	a look at this basic polynomial that follows an easy
01:13	to recognize pattern . Of course as I mentioned in
01:16	the last video , we don't really need to show
01:18	the coefficients of each term if they're just one like
01:20	we have here and the X to the zero term
01:23	is also just one . So we don't really need
01:25	to show that either , but I'm going to leave
01:27	it like this just for a minute to illustrate my
01:29	point . As you can see , this polynomial has
01:32	a term of every degree from zero up to four
01:35	. But do you remember that it was okay for
01:37	a polynomial to have missing terms . For example ,
01:40	we could have a slightly different polynomial that doesn't have
01:43	a third degree term . That makes it look like
01:45	the X cubed term gets skipped or is missing since
01:49	the pattern goes from X to the fourth , then
01:51	skips X cubed and goes to X squared and so
01:54	on . Well , just like there can be missing
01:57	terms in a polynomial . There can also be extra
02:00	terms like in this polynomial where the third degree term
02:03	has been duplicated , See how there's two terms that
02:06	have an X . Cubed variable part in this polynomial
02:09	. So this polynomial has no X cubed term which
02:12	is fine and this polynomial has just one X cubed
02:16	term which is fine . But this polynomial has two
02:19	X cubed terms which is also fine . But it's
02:23	more complicated than it needs to be . And whenever
02:26	you have terms like this terms that have the exact
02:28	same variable part , they can be combined into a
02:31	single term to do that . You just add the
02:34	number parts and you keep the variable part the same
02:36	. So one x cubed plus one x cubed combined
02:40	to form two x cubed . What we just did
02:43	there is called combining like terms like terms are terms
02:47	that have exactly the same variable part . But why
02:50	can we combine them ? Well to understand that I
02:53	like to pretend that the variable parts of a polynomial
02:56	terms are fruit . Yes . You heard me fruit
03:00	for example have a look at this polynomial . But
03:03	let's substitute a different kind of fruit for each different
03:07	variable part . Let's change X cubed to apple's X
03:10	. Squared to oranges and just plain X . Two
03:13	bananas . If we do that , what would this
03:16	new fruit polynomial be telling us ? Well this first
03:19	term represents two apples . The next term is four
03:22	oranges . The next term is three oranges and the
03:25	last term is five bananas . And these are all
03:28	being added together . So that raises the question .
03:32	What do you get when you add to apple's to
03:34	four oranges ? Well you get two apples and four
03:37	oranges . Since they're different fruit , you can't combine
03:39	them well , unless you have a blender that is
03:48	. Ah But what about the middle two terms ?
03:51	What do we get if we add four oranges and
03:53	three oranges ? That's easy . Seven oranges . And
03:56	that means that we can combine these two terms into
03:59	a single term which makes our fruit polynomial simpler .
04:02	Now , do you see why the variable parts of
04:04	a term have to be exactly the same in order
04:07	to combine them ? If the variable parts are different
04:10	, like X cubed and X squared , then they
04:12	represent different things . So we can't group them into
04:15	a single term . The way that we can ,
04:17	if the variable parts are the same . The mathematical
04:20	reason that it works that way has to do with
04:22	something called the distributive property , which is the subject
04:25	of a whole other video . All right . So
04:28	if two terms in a polynomial have exactly the same
04:31	variable part , then we call them like terms and
04:34	we can combine them into a single term to simplify
04:37	the polynomial . And to help you get better at
04:39	identifying like terms . Let's play a little game called
04:42	like terms or not ? Like terms . The first
04:46	pair of terms will consider is two X and three
04:48	X . Are they like terms ? Yep . The
04:51	variable part of both terms is the same X .
04:54	So we can combine them into a single term .
04:57	We do that by adding the number of parts and
04:59	keeping the variable part the same two plus three is
05:02	five . So the combined term is five X .
05:05	Next up , we have four X and five Y
05:08	. Are these like terms ? Well , they're both
05:11	first degree terms . But since the variables are different
05:13	letters , they are not like terms . That means
05:16	we can't combine them . Okay but what about these
05:19	terms ? Two X squared and negative seven X squared
05:24	. Well the variable part in both is exactly the
05:26	same . It's X squared . So yes , these
05:29	are like terms and we can combine them . Notice
05:32	that one of the terms is negative . So when
05:34	we add the number parts will end up with negative
05:36	five . So these combine two negative five X squared
05:41	. Our next pair of terms is four X squared
05:44	and six X cubed . Are these like terms ,
05:47	nope , even though the variable is X . In
05:50	both cases the exponents are different . So the variable
05:53	parts are not the same . Next we have negative
05:57	five X , Y and eight Y X . Are
06:00	these like terms ? Well at first glance you might
06:03	think that the variable parts of these terms are different
06:05	because the X and the Y are in a different
06:08	order . But remember multiplication has the community of property
06:12	. So the order doesn't matter . X . Y
06:15	is the same as Y . X . So we
06:17	can rewrite them so that they look the same to
06:20	there . Now we can add the number parts negative
06:23	. Five plus eight is three . So we wind
06:25	up with the single term three X . Y .
06:29	Last we have five X squared Y . And five
06:32	Y squared X . Now be careful with this one
06:34	. You might think that it's like the last one
06:36	where the terms are just in a different order .
06:38	But look closely in the first term the X .
06:41	Is being squared , but in the second term the
06:44	Y . Is being squared . That means even if
06:46	we switch the order , the exponents move with the
06:49	variables . So the variable parts are not the same
06:52	, which means these are not like terms . All
06:55	right . Now that you've had some practice identifying like
06:58	terms , let's look at some complicated polynomial is that
07:01	we can simplify by combining any like terms that we
07:04	find . Here's our first example X squared plus six
07:08	X minus x plus 10 . Do you see any
07:11	terms that have the same variable part ? Yep ?
07:14	These two terms in the middle both have the variable
07:17	X . So we can combine them six x minus
07:20	X would just give us five X since six minus
07:23	one is five . Remember if you don't see a
07:26	number part in the term , then it's just one
07:28	. So this polynomial started with four terms but simplified
07:32	to three terms X squared plus five X plus 10
07:37	. Let's try this . 1 16 -2 . x
07:40	cubed plus four X -10 . In this polynomial .
07:45	We have a third degree term , a first degree
07:47	term . And to constant terms are those constant terms
07:51	, like terms ? Absolutely . They're both just numbers
07:55	and don't really have a variable part so we can
07:57	combine them easily . This term is positive 16 and
08:01	this term is negative 10 . So if you add
08:03	them together , you end up with positive six .
08:06	Remember it's best to think of all terms in a
08:08	polynomial as being added , but they can have coefficients
08:12	that are either positive or negative . That's why this
08:14	negative sign stays here with the two X cubed term
08:17	because its a negative term . So this polynomial is
08:21	now as simple as it can be since there's no
08:23	other like terms ready for an even more complicated polynomial
08:29	. Three X squared plus 10 minus three X plus
08:33	five X squared minus four plus x . This polynomial
08:38	has six terms . And when you get a long
08:40	polynomial like this , the first thing to do is
08:43	look to see if any of the terms are like
08:45	terms . So you can combine them well right away
08:49	. You may notice that there's two constant terms in
08:51	this polynomial positive 10 and negative four . So let's
08:54	start by combining them into a single constant term positive
08:58	six since 10 minus four equals six . Next we
09:02	see that there are also two first degree terms negative
09:05	three X . And positive ex Those are like terms
09:09	. So we can combine them negative three X plus
09:12	one X . Gives us negative two X . Last
09:15	we see that there's also two different terms that have
09:17	the variable part X squared . So we can combine
09:20	them to three X squared plus five X squared ,
09:24	gives us eight X squared . So our polynomial started
09:27	out with six terms . But we were able to
09:30	simplify it to just three terms . Eight X squared
09:33	minus two X plus six . That almost made algebra
09:37	seem fun , didn't it ? All right . So
09:40	now you know how to simplify polynomial by identifying and
09:44	combining like terms . It can sometimes be a little
09:47	tricky since complicated paul . No meals may have many
09:50	different terms that are not necessarily in order by their
09:53	degree . That means you may need to do some
09:57	rearranging as you look for terms that you can combine
10:00	. I like to look for pairs that I can
10:01	combine and then once I combine them into a single
10:05	term and my simplified polynomial , I crossed them off
10:08	in the original polynomial . So I know that I've
10:10	already taken care of them . Any terms that can't
10:13	be combined . Just come down into the simplified polynomial
10:16	as is Oh . And to make things easier .
10:19	Don't forget to treat each of the terms as either
10:21	positive or negative depending on the sign right in front
10:24	of it . So that's how you simplify polynomial and
10:28	now that you know what to do , it's important
10:30	to practice simplifying some polynomial is on your own so
10:32	that you really understand it as always . Thanks for
10:35	watching Math Antics and I'll see you next time learn
10:39	more at Math Antics dot com .

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