Algebra

More examples of factoring by grouping | Algebra I | Khan Academy

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In this presentation, I want to focus on more techniques that are used To analyze polynomials into their factors Specifically, I want to focus on quadratic equations that do not Its coefficient is 1 For example, if I wanted to analyze 4x ^ 2 + 25x – 21 Everything we’ve focused on lately, or all of the quadratic equations We focused on it recently, it contained either 1 or -1 places This is 4 And now we have these 4 So what I’m going to teach you is a technique called Factor analysis by grouping It is more comprehensive than what it has We learned it before, but it is also brilliantly deceptive Somewhat, you’ll get old when you learn The quadratic formula, because the quadratic formula Slightly simpler But that’s how it goes I will teach you the technique And then at the end of the show, I’ll show you The reason for its success So what we’re going to need to do here is we’re going to think about Two numbers, a and b, with a x b = 4 X-21 So a x b = 4 x -21 Equivalent to -84 These same numbers, a and b, must Their total should be 25 Let me be clear This is 25, so their sum should be 25 From here came the 4 So we go, 4 x -21 It-21 So what are these two numbers? Well, we have to look at factors -84 Once again, one of them will Be positive The other one will be negative, because The product of both is negative So let’s think about the different factors That might work 4 and 21 seem exciting, but when you get them together You get -17 Or if we have -4 and 21, we’ll get positive 17 This did not work Let’s try other ingredients 1 and 84, they are very far apart when we take the difference between them Because that’s what we’re going to do, then One was negative and one positive They are very far 2 and 42 Once again, far away from each other -2 + 42 = 40 2 + -42 = -40 – two far apart 3 and – let’s see, 84 ÷ 3–8 ÷ 3 = 2 2 x 3 = 6 8 – 6 = 2 We go down the 4 = 8 exactly So 3 and 28 This seems very interesting 3 and 28 And remember, one of them must be negative So if we have -3 + 28, that’s equal to 25 Now, we have created the two numbers But it was not a simple process We did when this was 1 or -1 What we will do now is break down this phrase We will break it down to – We will break it down into positive 28x – 3x We will segment this phrase That phrase is this phrase And of course, we have -21 here, and we have 4x ^ 2 here Now, you will probably say, how did you choose the 28 to get here F-3 to move you here? In fact, this does not matter The way I thought was 3, -3, and 21, or -21, they have the same common factors Specifically, they own co-factor 3 28 and 4 have the same common factors So I collected 28 on one side with the 4 You will see what I mean quickly If we compile this, then this expression becomes 4x ^ 2 + 28x Then this side, written in pink here, is + -3x – 21 I chose them again I combined -3 with 21, or -21 Because they both divide by 3 And I combined 28 with 4, because both Divide by 4 And now, in each of these groups, we did the analysis As much as possible So both of these statements accept division by 4x This orange phrase is equal to 4x x x – and 4x ^ 2 Divide by 4x and output x– + 28x ÷ 4x = 7 Now, this second phrase And remember, we analyze everything It can be analyzed according to its factors Well, each of these phrases is divisible by 3 or -3 So we extract factor-3 The phrase x becomes + 7 Now, maybe something will appear in front of you We have (x + 7) 4x + (x + 7) X -3 So we can get the x + 7 factor Perhaps this is not entirely clear Perhaps you are used to analyzing An entire binomial equation But you can consider this a Or if you have 4xa – 3a, you can Extract the coefficient a I can leave this signal negative . Let me erase this positive signal from here Because it’s -3, right? + -3 is -3 So what can we do here? We have (x + 7) x 4x We have (x + 7) x -3 Let’s extract the coefficient x + 7 So we get (x + 7) (4x – 3) – These are 3 By this we have analyzed our binomial statement Sorry, we analyzed the square equation using grouping We analyzed it into two binomial terms Let’s do another example of it, because it is More comprehensive But when you can do it it will be more fun So let’s suppose we want to analyze 6x ^ 2 + 7x + 1 In the same way We will find a x b which equals 1 x 6, i.e. Equals 6 And we find a + b which is 7 This is more direct What are they – well, the obvious thing is 1 and 6, right? 1 x 6 = 6 1 + 6 = 7 If we have a = 1 Or let me not put signs in front of them The numbers are 1 and 6 Now, we’re going to break it down into 1x and 6x But we’re going to collect it, so it’s on the side of something He shares the same factor with him So we’re going to get 6x ^ 2 here, + – and I will put the 6x first because 6 And 6 have a common factor And then we get 1x, right? 6x + 1x = 7x Thats all about it Their sum should be 7 And then we have +1 on here Now, in each of these groups, we can analyze As much as possible So let’s extract the 6x operator from the first set It becomes 6x x – 6x ÷ 6x = x 6x ÷ 6x = 1 Hence, the second group – will We get + here But in this second set, we have x + 1 Or we can write 1 x (x + 1) You can imagine that I extracted Factor 1 Now, I have 6x x (x + 1) + 1 x (x + 1) Well, we can get the x + 1 factor If you extract the factor x = 1, this equals (x + 1) x (6x + 1) I run Reverse distribution feature So I hope you did not find this very bad Now, I will explain why The success of this magical system The reason for his success Let me take an example I will use general expressions Let’s suppose I have (ax + b) (cx – in fact, I am I am afraid to use a and b I think this will confuse you, because I am Use a and b here They will not be the same So let me use completely different characters Let’s assume that I have (fx + g) (hx – i will use j Instead of i You will learn in the future why it is not preferred Use i as a variable How much will it equal? Well, it will be fx x hx, which is equal to fhx Then fx x j So + fjx Hence, we will get g x hx So + ghx Then g x j + gj Or if we add the two expressions in the middle, we get fh x (x + – Combine these two phrases – fj + ghx + gj Now, what did you do here? Well, remember, on all these issues when we have Coefficient other than 1 or -1, we are looking for Two numbers whose sum is equal to this, and their product is equal to Quotient this with this Well, here we have two numbers together – let’s say a = fj Let’s assume that a = fj That is a And b = gh So a + b = that The middle laboratory a + b = that middle parameter Then what is the quotient of a x b? a x b = fj x gh X gh We can rearrange these phrases, and multiply A set of phrases, so you can rewrite them as f × h x g x j All are equal Well, what is the product of fh x gj? This is equal to fh x gj Well, this is equal to the first parameter × Constant So a + b = the middle operand And a x b = the first parameter x Constant For this reason analysis using aggregation is considered Successful, or how can we find something? The value of a and b Now, I will end up with something Different, but until we make sure we have A good understanding of the analysis into factors What I want to do is to teach you to analyze things to their factors Completely It is an additional thing I would like to have a full presentation on it But I think, it might be somewhat Clear to you So let’s suppose we have – let’s take a good example here – Suppose we have x ^ 3 + 17x ^ 2- – 70x You will say directly, this is not a square equation I don’t know how to solve such a thing It contains x ^ 3 The first thing that you should realize is that every phrase Here, divide by x So let’s extract x as a common factor Or better, let’s extract -x If you think -x, it becomes -x x – -x ^ 3 ÷ x = x ^ 2 17x ^ 2 ÷ -x = -17x 70x ÷ -x- = positive 70 The x is deleted And now, we have something that might sound Familiar somewhat We have a typical square equation The coefficient in it is 1 We just have to find two numbers multiplied by Khu 70 Their sum is equal to 17 The numbers that will come directly to my mind H-10 and -7 We take the product of both, which is 70 And we add them together, so the result is -17 So this part is going to be (x – 10) X (x – 7) And of course, we have that -x The general idea here is to see if we have anything It can be analyzed according to its factors This leads us to a form that you may realize I hope you find this offer useful I want to repeat what I explained at the beginning this show I think it’s a beautiful trick, to say, that you can To analyze the things that contain Parameter 1 or -1 But to some extent, you will find faster ways To do this, especially with the quadratic formula In a short time

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