Algebra

Multiplying binomials and polynomials | Algebra Basics | Khan Academy

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In this demo I will multiply a set of Polynomials, and I hope this will give you Feeling confident when you have to hit it By yourselves Let’s start with a simple matter Suppose we want to multiply 2x x (4x – 5) Well, here we will use Direct distribution feature In fact, when we multiply all these polynomials All we use is Distribution feature for several times But let’s use the distribution feature here This is equal to (2x x 4x) + (2x x -5) Or you can say – 5 x 2x So we’ll say, -5 x 2x All I did was distribute the 2x The first phrase will equal – we can multiply Transactions– Remember that 2 x x 4 is equivalent – you can Rearrange the multiplication process– This is equivalent to 2 x 4 x x x x Equivalent to 8 x x ^ 2 And remember, when you multiply x ^ 1 x x ^ 1, we add the exponents I mean, you know x x x = x ^ 2 So the first expression is 8x ^ 2 And the second phrase, -5 x 2 = -10x Not bad Let’s do another one Suppose we have 9x ^ 3 x (3x ^ 2 – 2x + 7) So again, we will use Distribution feature here We’ll multiply 9x ^ 3 by These phrases So 9x ^ 3 x 3x ^ 2 I will write it this time As for the following questions, we will start doing this Mentally So this equals 9x ^ 3 x 3x ^ 2 And then we have – let me write that down Method– – (2x x 9x ^ 3) then + (7x 9x ^ 3) Sometimes I wrote 9x ^ 3 first, and sometimes it was We wrote it later because I wanted to put this on The negative sign is here But this does not change the arrangement that We hit it So how much does this first phrase get? 9 x 3 = 27, x x ^ – We can add up Foundations, we learned this in a lesson about the properties of foundations– This x ^ 5, and -2 x 9 = 18x ^ – We have x ^ 1 and x ^ 3– x ^ 4 + 7 x 9 = 63x ^ 3 So we ended up with Fifth degree polynomial Now let’s do one solution that contains multiplying the two-term expressions I will quickly show you what I mean You will see this A lot in algebra So let’s suppose we have (x – 3) (x + 2) In fact, I want to show you that everything we do here is Use the distribution feature So let me write it this way: x (x + 2) So let’s pretend this is just too many indeed You know, if we had a group of x, then this would be a number here So let’s distribute this to each of these variables This is equal to (x – 3) x x written in green, + (x – 3) x 2 written in green All we did was distribute the x – 3 This is a distribution property And remember, if I have a x (x + 2), then what Equal? This is equal to (a x x) + (a x 2) So here, you can see that when x – 3 is equivalent a, we distribute it Now we will use the distribution feature again In this case, we will now distribute x to x – 3 We will distribute 2 by x – 3 Perhaps you are used to seeing the x on the other side, however In any way, you will do a beating So this equals – I will continue to paint – This is equal to (x x x) – (3 x x) + (x x 2) – I take the trouble to keep it going By coloring it for you I think this will help you– – (3 x 2) All I did was x distribution and 2 distribution Soon you will get used to this We can do it in one step You are actually hitting every phrase here In each subsequent phrase, we will find faster ways In the future to do this But I really want to show you the idea here How much is this? This is equal to x ^ 2 This equals -3x This is equal to 2x And then this equals -6 So the result is x ^ 2 – 3 of something + 2 of something, and that equals 1 of that thing – x – 6 We hit this one Now, before we continue and solve another issue, I want to I tell you that you can do this mentally You do not have to go through all these steps I just wanted to show you that this is about Distribution feature The fastest way to do this, if we have (x – 3) (x + 2) You will actually hit every phrase Here with all of these phrases So you’re going to say, x this x that x And you get x ^ 2 Then we have x this x that 2, and this equals 2x Then we have -3 x, that’s equal to -3 x Then -3 x 2 Equivalent to -6 So when you hit again, you get x ^ 2 – x – 6 This needs some practice for Ki You really get used to it Now the other thing I want to do – and the main thing is It’s exactly the same way – but I’ll hit A double-edged phrase with a three-word term, and that’s what Many people find him tired But we’ll see, if you stay calm It is not so bad (3x + 2) (9x – 6x + 4) Now, how do you use the same method that we followed in Previous offer We can take (3x + 2) and distribute it to For each of these three phrases, we multiply 3x + 2 by each From these phrases, and then we distribute Each of these phrases has 3x + 2 It will take a long time, and in fact, you will not hit her this way But you will get the same answer that we will get When you have polynomials greater than this, the simplest method is that I can think of it to hit, which is how to do The process of multiplying long numbers We will write it this way 9x ^ 2 – 6x + 4 And we’re going to multiply that by 3x + 2 And what I imagine is, when you hit regular numbers You have the place of ones, the place of dozens And ranks of hundreds Here, we have the status of constants First class status, second class status And the status of the third degree, if present Indeed, it is present in this show So you have to put things in the right place let’s do it We’ll start from here, just hit it We do traditional beating 2 x 4 = 8 And you go to the place of ones, or the constants 2 x-6, 2 x-6 = – or 2 x -6x = -12x And we’ll put + here This was = 8 2 x 9x ^ 2 = 18x ^ 2, so we put this in Status x ^ 2 Now let’s move on to the 3x segment I will do it in purple, and as you can see it is different 3x x 4 = 12x, positive 12x 3x x -6x, how many do you have this? x x x = x ^ 2, so He will go here 3 x -6 = -18 And then finally, we have 3x x 9x, the x x x ^ 2 = x ^ 3 3 x 9 = 27 I wrote it x ^ 3 Once again, you will collect Similar phrases, you will get 8 There is no other constant, so we only have 8 -12x + 12x, these are deleted 18x ^ 2 – 18x ^ 2 is deleted, so we have more left Here 27x ^ 3 So this equals 27x ^ 3 + 8 And we’re done You can use this technique to multiply the term polynomial With the term binomial, the term polynomial x term polynomial, or in fact You know, you can get five phrases here A phrase from the fifth degree × a phrase from the fifth degree This will always work, as long as you keep things in Its appropriate position

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