Phân tích đa thức thành nhân tử bằng phương pháp nhóm hạng tử - Bài 8 - Toán 8 - Cô Chi (HAY NHẤT)

Hello everyone! In the previous lessons, we learned how to analyze the plural form of the word by the method of putting the word together and the method of using the plural form. Today, we will learn another method, which is to analyze the plural form of the word by the method of grouping the words. Let's go to the example part. Example 1 requires the analysis of the plural form of the word. x square minus 3x plus xy minus 3y Here, it is clear that we do not see that this is a plural form, or that we can put the plural form out. But when we look at x square minus 3x, we see that we can put x out. When we put x out, there will be x square minus x square minus 3. Continue. On this side, we have xy minus 3y, so we can put y out. When we put y out, we still have x square minus 3. Then we can put x square minus 3 out again. So we will do today's method, which is to group the words by putting two words together. We will group it into a group. Here, by x square minus 3x plus xy minus 3y After grouping, we will use two methods, which is to put the plural form or use the plural form. In this example, we will put the plural form. In the first group, we will put x out, so there will be x square minus 3. Plus here, I can put y out, so there will be x square minus 3. Then we continue to put x square minus 3 out, and in the box there is only x plus y left. So we have analyzed the above three forms into a plural form. But in addition, we can do the second method as follows. You can group, I call this the first, second, third and fourth. You can group the first and third words into a group, the second and fourth words into a group. Why did you group it like that? Now let's look at the second method. x square minus 3x plus xy minus 3y Now I will group the first word with the third word, which is x square. x square plus xy We note that when grouping, we always have to put the sign in front of that word. Here, xy, we will put the sign in front of it, plus. Not put the sign behind x square, but put the sign in front of xy. When you run out of squares, close the square and add. Plus with open square, we bring the sign in front of it. That is, minus 3x minus 3y into a group. Then I continue to put in the first group. What will I put out? Ah, put x out, right? So there will be x plus y in the square. In the second group, we will put minus 3 out, right? Put minus 3 out, it will be plus with minus 3. Minus with open square, x. Ah, x plus y. We note that leaving minus 3 here means losing this minus sign. Become a plus sign. Note the sign for her. I am not wrong in this part. Then we see that there is a common denominator, which is x plus y. Then I put x plus y out. So there will be x plus and minus 3 in the square. Or in other words, x minus 3. So when doing this exercise, we will have to use the idea group method. And you can use the group method like this or like this. As long as the results are correct and the same. And when doing other exercises, we need to be flexible. See if I can group ideas together. Can you put a common denominator when grouping or not? Or create a formula or not? Then we just group in. Example 2. Please think about how to group so that we can analyze these 3 formulas. 2xy plus 3z plus 6y plus xz. How do we group? Suppose now she will group two numbers next to each other in a group. Then 2xy plus 3z, we can not group, can not put a common denominator out. So I think I will group the numbers with z together. That is, the group of the second number with the fourth number in a group. The first and third numbers in a group, right? And now we will group together. By 2xy plus 6y plus 3z plus xz. Her letter z will be crossed like this. Because the letter z is quite similar to the combination of the original number. So she will write like this. In the first group, what can we put out? Give as much as you can. First of all, we will be able to put 2 out. As for the variable, we can put y. So there will be x plus 3 in the bracket. Continue. Plus z. Here, put z out, there will be 3 plus x in the bracket, right? 3 plus x is x plus 3. But now she still writes 3 plus x. But 3 plus x, we have to understand that it is x plus 3. And now I can put the variable x plus 3 out. At that time, there are only 2y plus z in the bracket. And this is the final result. Let's go to the second part. First, let's go to the third example. Example 3 requires us to calculate quickly. And there is the following formula. Here, we do not calculate specifically. Because 15x64, we multiply will take time. But I will use the method we learned today. That is, group, then use, set the common factor. Or use the formula to do. Here, I still group and set the common factor. So you guys look, we will group what and what. 15x64, we can group with. Ah, group with 36x15, right? Because we can get the common factor is 15 out. We will group 25x100 with 60x100. Because I can get 100 out. Now, we will group together. First, 15x64 will add up to 36x15. Or add and open. 25x100, then group with 100. Ah, 60x100, right? Must put it back. Already. Here, I set the common factor is 15 out. In the box, there will be 15x64, plus 36. This place set 100 out. In the box, there will be 25x60. Yes, 15 down. 64x36 is fine. Ah, get 100. So, after grouping, we can set the common factor is 100 out again. Right? 25x60, we get 85. So here, set the common factor is 100 out. In the box, there will be 85x15. But 85x15 is how much? Again, right, 100 more. So we will get the result is 100x100. Now, multiply 100x100, we will get how much? 1 and add 4 numbers, 0 in. So we get the result is 10,000. Go to the 4th example. The 4th example is quite long, but we do not worry. Now, we will analyze together. The 4th example requires you to analyze multiple factors. x4 minus 9x3 plus x2 minus 9x. Now, there are 3 people. They are Thai, Ha and An. What did you do? Please look at your homework and give me your comments. Did you do the homework correctly? First is Thai. What did he do? With x4 minus 9x3 plus x2 minus 9x, he got x out. To get x out, there will be x3 minus 9x2 plus x minus 9. Did Thai do it correctly? Yes, he did. Next is Ha. Ha is in the group. He is in the group. The group of the first letter with the second letter, the third letter and the fourth letter in a group. He got x4 minus 9x3 plus x2 minus 9x. Then, in the first group, he saw that there was a common factor of x3. He took x3 out. What is left here? x4 is equal to x3 minus x. There will be x minus 9. Here, you can see that there is a common factor of x. He took x out. There will be x minus 9 inside. Here, I see that there is a common factor of x minus 9. So he took x minus 9 out. And there is x3 plus x inside. Did Ha do it correctly? Yes, he did. Next is An. What did An do? An is also in the group, but he is a bit different from Ha. He is in the group of the first letter with the third letter in a group, the second letter with the fourth letter in a group. What did he get? x4 plus x2 minus x3 plus 9x. So why does he have this? Just now, she guided us. Right from the first lesson, she guided the group of letters that after all the brackets, we will add with the open bracket and we bring the mark in front of it. Right? And here, An has done a bit of work. He was able to take the minus sign out. Those of you who can take the minus sign out like An, it's good, but you have to do it right. If I do as she instructed, we will get plus with the open bracket minus 9x minus 3 minus 9x. Right? So when I bring the minus sign out, bring the minus sign out so that all the brackets inside have to be hidden, we will also get 9x minus 3 plus 9x. Do you understand this? If you do it quickly, please do it like this for her. Continue. Here, you see the plural is x. Take out x. Take out x, in this x4, we only have x. In x, take out x, there is only one. There is one, remember, not zero. In this place, we have to understand that x square is 1, so we will get x square with the open bracket x square plus 1. Continue. Here, I see the plural is 9x, so I bring 9x out, so inside the bracket will be x square plus 1. Then I see the plural is x square plus 1 again, so I bring it out, so I get x square plus 1 with the open bracket x square minus 9x. Up to this point, you will see that x square minus 9x with the plural is x. Then you bring it out, you get the final result, x square minus 9x with the plural is x square plus 1. So did An do it right? Yes, right? So all three of you did it right. However, why did An do it longer? Ha did it shorter than An a bit, and Thai did it shorter like this. And the result is not the same. So are all three of you right? I said that Thai and Ha did it right, but it was not complete. So if I score Ha and Thai, I will not give them enough points. Why? Because I said right from the start, when we analyze the plural form, we have to analyze all of our abilities. And here, it is clearly not all of our abilities. Why? We can group x square minus 9x into a group, to analyze further. Here, I will continue to make it clear for us. First is Thai's problem, and I will not copy the part that he has finished analyzing, but I will write it as it is, and we will understand that it is from the part he has finished analyzing. Here, I will group x square minus 9x into a group, minus 9x square, plus square root of x minus 9 into a group, right? What will I do next? Here, I will group x square minus 9x into a group. Here, I can put x square out, and note that there is still a square. If you put x square, there will be x minus 9, plus x minus 9 here. So what will we get? We continue to put x minus 9 out, equal to here, x square minus 9x, and what will be left in the group? It is still x square plus 1, right? There is nothing here, so I will understand that it is plus 1. So, I got the result just like Anh, and when I got here, Thai had enough points. Let's continue to Ha's problem. He got the result as x square minus 9x plus x, right? But clearly, we see that x square minus 3 plus x, we have a common factor, which is x, right? So, I continue to put x out, then there will be x square, x square plus 1. Here, Ha just got enough points. Can you guys still hold it? Let's move on to the last few exercises. The first exercise is the 47th exercise in the 22nd textbook. It requires us to analyze the multiplication of factors, and we definitely have to use the group method, then use the common factor, or use the equation. Now, let's do part A together. Part A, I think, is x square minus x, plus x minus y. Now, how are we going to group now? We can group x square with x minus y in a group, x minus y in a group, right? Because here I can put the factor x out. When x is put out, inside the box will be x minus y. This side has x minus y, so we can do it. Or you can do it another way, like this. You group x square with x in a group, and minus xy with minus y in a group. When x is put out, we will get x minus x plus 1. The second group, we will get minus y minus x plus 1. And we will get the result like the first way we did. And I will only show the first way. And the second way, let's use what you just suggested and do it yourself. Now, we will group together in the first way, that is, groups of two numbers next to each other. When x is put out, inside the box will be x minus y. This side has x minus y. So, in short, we will get x minus y minus x plus 1. Note number 1, don't forget. Continue with part b. In part b, I give xz plus iz minus 5 times x plus y. Here, people have grouped 5 times x plus y already, right? Here, if we group the first two numbers, we will have the common factor z, right? When z is put out, inside the box will be x plus y. Here also has x plus y. So, we will finish the lesson. Now, let's group xz plus iz minus 5 times x plus y. And by putting z out, inside the box will be x plus x plus y minus 5 times x plus y. Here, we can put x plus y out. And inside the box, there is only z minus 5. Part c. In part c, I give 3xy minus 3xy minus 5x plus 5y. How do we do this? Please pay attention. Here, the number sign is 3. The number sign is 3. And here, the number sign is 5. The number sign is 5. So, we will think that we group two numbers together into one group, right? Now, let's group 3xy minus 3xy into one group. When we finish the lesson, we will still do it by adding with the open box so that we don't get it wrong. minus 5x plus 5y. Now, here. What did we put out in the first group? The maximum. We put 3 out, right? And we put the minus sign out. So, we put 3x. When we put 3x out, what is left in the box? 3xy is left as minus 3xy. When we put 3x out, we still have y. Here. I will put the minus sign out. Why did I put the minus sign out but not the 5? Because when we put the 5, in the box, we still have minus x plus y. In the previous lesson, she already guided us. Minus x plus y is the opposite of minus y. So, when we put the minus sign out, we still have minus x plus y. So, we will take a step back. Let's put the minus sign out. When we put the minus sign out, we still have x. When we put the minus sign out, the previous sign automatically changes to minus. You have to pay attention to the sign. First, we will put minus x plus y out. So, in the box, we still have 3x minus 5. So, we have finished the first lesson. The second lesson is the lesson 48 in the textbook chapter 22. The lesson still requires us to analyze multiple factors. Now, we will do this lesson together. First, x square plus 4x minus y square plus 4. If we still do the first lesson, that is, we group 2 factors into 1 group, we see that when we group them in, we cannot put the common factor out. Right? So, we have to think of another way, which is to use the equation. Here, you have to be careful. You have to realize that x square plus 4x plus 4 is an equation. Do you realize that? Now, we need to do more exercises to analyze multiple factors using the equation method. If you realize it, now we will apply it to do this part. So, we will group x square plus 4x plus 4 into 1 group. There, mark it. And the rest, minus y square. I will write it next time. Minus y square. Now, let's use the equation. x square plus 4x plus 4 is familiar, right? What is it equal to? It is equal to x plus 2. All of them are 2. Here, if you are not familiar with it, you don't need to follow the steps as in the previous lesson. Let's make a sum to determine how much is A and how much is B. Obviously, A square is equal to x square, right? B square is equal to 4. So, A is equal to x, and B is equal to 2. Let's check if it is correct. Obviously, 4x is equal to 2 multiplied by 2 multiplied by x, right? So, it is exactly equal to x plus 2 all of them, minus y square. Now, let's use the third equation to analyze further. A square minus B square is A minus B multiplied by A plus B. So, we will get x plus 2 multiplied by x plus 2 multiplied by y. Now, let's check B. B is equal to 3x square multiplied by 6xy multiplied by 3y square minus 3z square. So, how do we solve this problem? First, we see that all of them have the common factor 3, right? So, we will put the common factor 3 out. When we put it out, there will only be x square plus 2xy plus y square minus z square. Now, what will we do next? Here, we have already analyzed the common factor, but it is not enough. We have to do all the possibilities. So, here, x plus 2xy plus y square is the common factor, right? So, we will continue. We will get 3 multiplied by x square plus 2xy plus y square in a group, right? And here, there is a circle. So, the outside will automatically become a square. And we will subtract z square. Here, we will continue. x square plus 2xy plus y square is x plus x plus y, all of them, minus z square. And here, it is still a square. Now, I will analyze the common factor by using the third method. We will get x plus y minus z minus x plus y plus z. There should be a square outside, but because it is the common factor, the square is not important. I have removed the square. The last part, the part c. The part c is for x square minus 2xy plus y square minus z square plus 2zt minus t square. Ah, here. Here, we have to realize x square minus 2xy plus y square is a common factor, right? Let's continue to this side. minus z square plus 2zt minus t square is also a common factor. However, it is a bit different. Because here, the coefficient of minus x square of z square and t square is a minus sign, right? And our common factor is a square plus b square. That is, both are plus signs. However, when we put the minus sign outside, we will get minus z square minus 2zt plus t square. So that is another common factor, right? Remember, when the square has the same sign, it can be a common factor. The first or second common factor. Now I will go to the group x square minus 2xy plus y square in a group. And here, I will do it quickly by putting the minus sign out. If I put the minus sign out, what will be left in the group? z square minus 2zt plus t square. Especially, I have to hide the minus sign here, plus sign here, minus sign here, when I put the minus sign outside. And here we will get x square minus y square. It's easy. We will get z minus z minus t. After that, we will continue to use the third common factor. We will get x minus y minus minus z minus t. Now I still put the minus sign outside so that we get used to it. And after that, I will skip this step. We will delete the minus sign and write the result. The second is x minus y plus z minus t. We don't need to hide the minus sign so we will write the square sign. Because we don't need to hide the minus sign in the previous step. If we hide the minus sign in the previous step, we will be afraid that you will be wrong. And here, if we hide the minus sign, we will change it to minus z plus t. So we will get x minus y minus z plus t times the square of x minus y plus z minus t. So we are done. The third exercise is 49 in the textbook page 22. It requires us to calculate quickly. And here I give you 2 parts. I think we can do part A by ourselves because you will get used to how to group to get more common factors. Obviously, here we see that the second and third factors are 7.5, right? The first and fourth factors are 37.5. So we know that we will group together, right? Now I will do it below. Part A. Let's review the problem. 37.5 times 6.5 minus 7.5 times 3.4 minus 6.6 times 7.5 plus 3.5 times 37.5 Here we will group. First, with the fourth factor, we will group 37.5 times 6.5. The first factor is plus plus 3.5 times 37.5 minus 7.5. Now we will group the second and third factors together. Obviously, the first and third factors are all negative. So I will put the negative out, okay? Put the negative out. What will be left in the group? 7.5 times 3.4 The negative here will turn into a positive. And we get 6.6 times 7.5. Up to here, we will get 7.5 minus 6.5 plus 3.5 minus 7.5 minus 3.4 plus 6.6 We see that 6.5 plus 3.5 is 10, right? 10 minus 7.5 times 3.4 plus 6.6 is also 10. So I put 10 out again. What will be left in the group? 3.7 minus 7.5 So we will get 10 times 3.7 minus 7.5 Ah, only 30 left. So the final result will be 300, right? 10 plus 0. Let's continue with part B. I think it will be a little more difficult for you to understand. But I think you will get it right away. 45 plus 40 minus 15 plus 80 times 45 We have to see that there are 3 squares here. 45 squares 40 squares minus 15 squares Then we will think about part 2. In part 2, I will analyze the form A plus B of all squares minus Dc of all squares so that later I can use the third derivative, right? So here it is also the same. 45 squares plus 40 squares Here is 80 times 45. So 80 we have to write it into 2 minus 40. From there we have the derivative, right? So I will group first 45 squares plus 40 squares plus 80 times 45 There, I bring the first symbol minus 15 squares Then we will use the derivative 80 is 2 times 40 times 45 minus 15 squares This derivative is a bit different because we are writing 2AB at the bottom. This is A square plus B square plus 2AB So we will still understand A square plus 2AB plus B square So we will get 45 plus 40 all squares minus 15 squares Do you understand? Then we use the third derivative. It's simple, right? 45 plus 40 minus 15 times 45 plus 40 plus 15 45 plus 40 we get 85. 85 minus 15 is 70 45 plus 40 we get 85. 85 plus 15 is 100 So we will get 70 plus 2 zeros or 7,000 The last problem is problem 50 in chapter 23 Find X Here I give 2 parts We have to do it by putting the common factor out Analyze the equation of the factor We have to put the common factor out to get the equation It is the form A times B equals 0 And we will solve this form by equal to A equals 0 or B equals 0 The decimal mark is the decimal mark Or if you write it another way of class 6 and class 7 we will divide it into 2 cases Case 1 is A equals 0 Case 2 is B equals 0 Can you get it? Now we will do each part together to make it clearer In part A, I give X times X minus 2 plus X minus 2 Now let's analyze 3 equations of the factor We can see what is the common factor It is X minus 2 So here I will understand that X minus 2 is equal to 1 I will write one more step You don't need to write one more step I know that for those of you who are not clear, you can understand our exercise better Now, do you see the common factor is X minus 2? Put X minus 2 out What is inside? It is X plus 1 Here solve as I just instructed A equals 0 X minus 2 is equal to 0 Or X plus 1 is equal to 0 Here I will solve as X equals 2 or X equals X minus 1 And here we will conclude So X minus 1 X equals 2 That's it for part A Is it easy? Part B is the same Let's go to part B I give 5X times X minus 3 minus X plus 3 equals 0 Here I have to remember A minus B is equal to minus B minus A Is it right? Ah, we don't need this Actually, we can make it easier We just need to put the letter out of the bracket Here we don't need to know what 3A minus B is to make it easier This is equal to 5X times X minus 3 Now put the letter out What do we need? Put the letter out or X plus X minus 3 Is it easier? Then we do like part A Put the letter out or 5X minus minus 1 We don't need to know 5X 5X minus 1 is right Equal 2 cases I will show you how to do it like part 6 and 7 I will show you how to do it In case 1 How to do it? In case 2 X minus 3 is equal to 0 We put the letter out X minus 3 is equal to 0 In case 2 We put the letter out 5X minus 1 is equal to 0 Here 5X is equal to 1 And X is equal to 1 by 5 Here we conclude X is equal to 3 X is equal to 1 by 5 Remember to conclude In this lesson You have learned how to analyze multiple real numbers by putting real numbers together using linear functions and groups of real numbers The exercise in the textbook has been solved You can apply it and make more exercises in the textbook It is exercise 31, 32, 33, page 10 Today's lesson is over If you have any questions please leave a comment I will answer as soon as possible See you in the next lesson