Algebra trick for multiplying polynomials - Free Educational videos for Students in K-12

Algebra trick for multiplying polynomials - By tecmath

Transcript


00:00	Good day and welcome to the Tech Math channel we're
00:02	going to have to look at in this video is
00:04	a way of multiplying any types of polynomial together ,
00:07	but especially really , really big ones and are more
00:09	difficult sorts of ones . So this is the fourth
00:12	video I've made where we're looking at the multiplying these
00:14	types of algebraic expression . So I'll put some links
00:16	up to the earlier ones . They're probably really worth
00:19	seeing if you haven't seen them yet . But so
00:22	we're gonna be multiplying some really , really big polynomial
00:25	in this one . So I'll give you an example
00:27	here , um , to get fairly rapid answers without
00:31	doing huge amounts of writing and equals and equals and
00:34	equals and equals . So the two polynomial are going
00:38	to get you multiplying together as follows to execute and
00:46	X squared plus three X plus four . We're going
00:52	to multiply this boy four X cubed plus two X
00:59	squared class two X Plus three . Okay . And
01:07	you just set this up like a fairly standard mathematical
01:10	calculation . But um there is a fair degree of
01:15	difference between this and a standard modification now to help
01:18	us to visualize what we're doing . To put a
01:21	bunch of dots up here just to represent each one
01:23	of these . So this okay , so that we
01:26	got 44 part polynomial expression here . So there's four
01:31	dots here to represent that and we've got a four
01:33	part of here . There's four dots to represent that
01:36	. Okay , so the foot to get this first
01:39	part of our answer , what we're gonna do is
01:40	we're going to multiply these guys together here . Okay
01:44	. This will give us this part of our answer
01:46	here . So four threes 12 . Okay , positive
01:50	. 12 . Okay . The next part of our
01:54	answer to get this , we're going to multiply this
01:58	one and this one and this one and this one
02:00	. Okay , so three times three X . And
02:03	four times two weeks we're going to add these together
02:05	. So three times three X which is nine X
02:08	. And four times two X . Which is a
02:09	dex . We add these together . Nine X plus
02:12	paid X is 17 X . Okay to get the
02:17	next part of our answer , you're going to see
02:19	this pattern . Hopefully you see this pattern . It's
02:22	a pretty easy one to pick up . We're going
02:24	to multiply this one , this one , this one
02:26	, this one and this one and this one and
02:29	we're gonna add our answers together . This is going
02:31	to give us the X squared part of our answer
02:34	. Okay so three times X 33 times X squared
02:39	is three X squared three x times two X six
02:43	X squared and four times two X squared is eight
02:46	X squared . So we're gonna add these together .
02:48	So we got three plus six plus eight . So
02:53	three plus six plus eight is 17 X squared .
03:00	Okay that's a positive that one earlier too . I
03:01	should be putting the positive there as we're going along
03:04	apologize for that . So the next part of our
03:07	answer is going to be the X cubed part of
03:09	our answer . Okay So we're going to be multiplying
03:11	this one now by this one we're gonna be multiplied
03:15	this one by this one this one by this one
03:17	and this one by this one . Okay . Yes
03:21	his pattern here . Okay so this will give us
03:24	her ex Cube part of our answer . Okay so
03:27	I'll even put that x cubed there now . So
03:30	three Times two x cubed is six x cubed two
03:35	x times x squared is two x cubed . Two
03:38	X bye two X squared times three X . Is
03:43	six execute and four executed by four is 16 X
03:47	cube . So There's an X . Cube . So
03:50	we have six plus so six plus two which is
03:57	eight plus six which is 14 . I was 16
04:02	which is 30 . Okay so that's 30 execute .
04:11	Okay we'll go to the next part of our answer
04:14	right now . This bit we're moving now across so
04:17	we're going to be now multiplying this one and this
04:19	one this one and this one mhm This one and
04:24	this one so let's see how we go here .
04:27	So first off what we have is this is gonna
04:30	be by the way that's a positive our last one
04:32	this is gonna be our X . To the power
04:33	of four answer because So two x times two x
04:38	cubed . Yes . Ah Four X . 44 X
04:43	. To the power of four . These two multiplied
04:46	by together is gonna be two excellent power four .
04:48	And these guys multiplied together are going to be 12
04:52	X . to the power of four . So four
04:55	Plus two which is six plus 12 is I think
05:01	. Okay . Doing pretty well . Almost done .
05:08	The next two we're gonna get getting and that's a
05:09	positive . I'll put that in there . It's a
05:12	positive . The next thing we're gonna do are these
05:16	two ? Okay , So This is going to give
05:19	us her X . to the power of five answer
05:23	. Okay , so two times 2 is four ,
05:28	four times one is forced at four plus four .
05:33	And the next one we're going to get finally is
05:36	our last part of our answer . We're going to
05:39	be multiplying these two guys together . This is going
05:41	to give us our X . To the power of
05:43	six answer . It's a positive . So X .
05:46	The power of six . Okay . two times 4
05:51	. Is that there's our answer ? That's pretty cool
05:55	. Right ? I mean that's one huge polynomial expression
05:58	being solved and that that is the correct . But
06:01	let's let's check to see this is correct . Okay
06:04	so to do this we're gonna substitute in one into
06:08	our X . Okay . Here so what we're gonna
06:10	do is two plus one plus three plus four plus
06:13	two plus one is three plus 36 plus four .
06:18	His team four plus 26 plus two is eight plus
06:24	three is 11 . We have to multiply these together
06:26	. We get 110 . Okay ? And then we
06:31	start adding these guys up together . Okay this is
06:34	the coefficients here . So eight plus eight . It's
06:37	16 plus I'd . His 34 plus 30 is 64
06:44	plus 70 17 . I apologize . is 81 plus
06:50	another 17 is 98 plus 12 is 110 . So
06:55	our answer is correct . Okay , wow , that's
07:00	pretty that's a pretty big answer there . So hopefully
07:04	you got the idea of this . So we're just
07:06	just to recap on that one , we're going to
07:08	get these the way you do , this is your
07:10	first multiply these two expressions to get this part of
07:13	the answer . These two to get this part ,
07:15	these two , these three now to get this one
07:20	, these guys here to get this one , these
07:25	guys here to get this one , these guys here
07:28	to get this and then there's to get this .
07:30	Okay , so it's a fairly standard sort of thing
07:34	. Okay . It's a great little method though ,
07:36	because this otherwise would take you a long , long
07:39	time to work out . So what about we try
07:43	one more one that's slightly different to keep those up
07:48	there , that particular diagram , because I think it's
07:51	quite handy just to express what's going on here .
07:55	So what about I do a bit of a different
07:57	one ? Say I was to do forex , cute
08:05	plus two X squared plus three X plus four .
08:13	And what I'm gonna do is I'm gonna multiply this
08:17	. Bye five X cubed plus four X . And
08:27	I'm just going to multiply these guys together . Okay
08:30	, now what you might realize straight away is we
08:35	have a bit of an issue here , we have
08:36	a few gaps . We don't have any X squared
08:38	value here . We don't have a unit's value here
08:41	. So there's a bit of a trick for doing
08:43	these so we're gonna get this four X . We're
08:45	going to move it . Okay , so keep that
08:47	forex in mind . We have I'm going to move
08:50	it over here so it lines up so it lines
08:54	up With an exhale . So four x lines up
08:58	here and for these other ones we have no X
09:01	. Squared value . So I'm just gonna put a
09:02	zero there , we have no units value . I'm
09:05	just gonna put a zero there . Okay ? And
09:08	we can start now doing what we were doing before
09:11	. Okay So we multiply these guys to get our
09:14	units multiply these guys together , we actually get Fair
09:20	result because Nothing Times four is nothing . Okay ?
09:24	So we do the next part , we're going to
09:27	end up with ex answers which is going to be
09:32	someone actually get rid of the zero , I don't
09:34	think we need to leave it there . So we're
09:37	gonna end up to get an extra value . We're
09:38	gonna go uh three X times nothing which is nothing
09:41	in four X times four which is 16 X .
09:44	So our answer there is 16 X . To get
09:47	our next part which is our X squared part .
09:53	We're going to multiply these three guys . Okay ,
09:57	so the zero times two X squared is nothing and
10:01	zero times four is nothing . So are only part
10:04	that actually matters is three X times for X ,
10:07	which is 12 X squared . The kind that's a
10:12	positive to now get this next part . We're gonna
10:15	multiply these ones or this . Okay . So what
10:22	we're gonna do now is nothing times forex is nothing
10:28	four x times two x squared is eight to the
10:33	X to the power of three . This one is
10:35	going to give us nothing and this one is going
10:36	to give us 20 X to the power of three
10:38	. So 20 plus eight is 28 X to the
10:44	power of three positive . Let's now go to the
10:48	next one which is where we're going to be multiplying
10:49	this , this , this and this , this and
10:52	this . Sorry , let's do that . Uh ,
10:56	four x 4 because there's gonna be an X four
10:59	. We don't even need to we can just keep
11:01	our keep it , you know , not even worry
11:03	about saying export . We can just go four by
11:04	4 . 16 , two is nothing and three is
11:08	15 , 16 plus 15 is 31 positive . There
11:14	we go to the next one , which is going
11:15	to be an X five value four . We're gonna
11:19	be multiplying these guys . So four times nothing .
11:22	There's nothing . two times 5 is 10 And then
11:27	we're gonna finally get our last answer . Multiplying these
11:30	guys , which has an extra six value four times
11:34	5 is 20 . Okay , And that's our answer
11:38	. It's a pretty big answer . Let's check out
11:41	pretty big answer . So we can go with a
11:44	value of one into this . So we're going to
11:46	end up with four plus two plus three plus four
11:48	. So four plus two is six plus three is
11:50	nine plus four is 30 5-plus 4 is nine .
11:57	We multiply these together 13 times nine Is 117 .
12:04	So 20 plus 10 is 30 Plus 31 is 61
12:10	Plus , 28 , plus 12 is 101 Plus 16
12:17	is 117 . So our answer is correct . Okay
12:23	, so hopefully you like these . The next video
12:26	is we're gonna be having a look at this .
12:27	We're gonna be having a look at how to divide
12:30	polynomial is . Okay , so this video will be
12:33	up fairly soon or maybe when you're watching it type
12:35	already . Okay , so I hope to see you
12:37	next time . We'll see you then . Bye .

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