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More examples of factoring by grouping Algebra I Khan Academy.mp3 | In this video, I want to focus on a few more techniques for factoring polynomials. And in particular, I want to focus on quadratics that don't have a 1 as a leading coefficient. For example, if I wanted to factor 4x squared plus 25x minus 21. Everything we've factored so far, or all of the quadratics we've factored so far, had either a 1 or a negative 1 where this 4 is sitting. All of a sudden now we have this 4 here. So what I'm going to teach you is a technique called factoring by grouping. And it's a little bit more involved than what we've learned before, but it's a neat trick. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Everything we've factored so far, or all of the quadratics we've factored so far, had either a 1 or a negative 1 where this 4 is sitting. All of a sudden now we have this 4 here. So what I'm going to teach you is a technique called factoring by grouping. And it's a little bit more involved than what we've learned before, but it's a neat trick. But to some degree, it'll become obsolete once you learn the quadratic formula, because frankly, the quadratic formula is a lot easier. But this is how it goes. I'll show you the technique, and then at the end of this video, I'll actually show you why it works. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | And it's a little bit more involved than what we've learned before, but it's a neat trick. But to some degree, it'll become obsolete once you learn the quadratic formula, because frankly, the quadratic formula is a lot easier. But this is how it goes. I'll show you the technique, and then at the end of this video, I'll actually show you why it works. So what we need to do here is we need to think of two numbers. We're going to think of two numbers, a and b, where a times b is equal to 4 times negative 21. So a times b is going to be equal to 4 times negative 21, which is equal to negative 84. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | I'll show you the technique, and then at the end of this video, I'll actually show you why it works. So what we need to do here is we need to think of two numbers. We're going to think of two numbers, a and b, where a times b is equal to 4 times negative 21. So a times b is going to be equal to 4 times negative 21, which is equal to negative 84. And those same two numbers, a and b, a plus b, need to be equal to 25. They need to be equal to 25. Let me be very clear. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | So a times b is going to be equal to 4 times negative 21, which is equal to negative 84. And those same two numbers, a and b, a plus b, need to be equal to 25. They need to be equal to 25. Let me be very clear. This is the 25, so they need to be equal to 25. This is where the 4 is, so they need to be equal to 4 times negative 21. That's a negative 21. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Let me be very clear. This is the 25, so they need to be equal to 25. This is where the 4 is, so they need to be equal to 4 times negative 21. That's a negative 21. So what two numbers are there that would do this? Well, we have to look at the factors of negative 84. And once again, one of these are going to have to be positive. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | That's a negative 21. So what two numbers are there that would do this? Well, we have to look at the factors of negative 84. And once again, one of these are going to have to be positive. The other ones are going to have to be negative, because their product is negative. So let's think about the different factors that might work. 4 and negative 21 look tantalizing, but when you add them, you get negative 17. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | And once again, one of these are going to have to be positive. The other ones are going to have to be negative, because their product is negative. So let's think about the different factors that might work. 4 and negative 21 look tantalizing, but when you add them, you get negative 17. Or if you had negative 4 and 21, you'd get positive 17. It doesn't work. Let's try some other combinations. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | 4 and negative 21 look tantalizing, but when you add them, you get negative 17. Or if you had negative 4 and 21, you'd get positive 17. It doesn't work. Let's try some other combinations. 1 and 84, too far apart when you take their difference, because that's essentially what you're going to do if one is negative and one is positive. Too far apart. Let's see. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Let's try some other combinations. 1 and 84, too far apart when you take their difference, because that's essentially what you're going to do if one is negative and one is positive. Too far apart. Let's see. You could do 2 and 42. Once again, too far apart. Negative 2 plus 42 is 40. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Let's see. You could do 2 and 42. Once again, too far apart. Negative 2 plus 42 is 40. 2 plus negative 2 is negative 40. Too far apart. 3 goes into 84. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Negative 2 plus 42 is 40. 2 plus negative 2 is negative 40. Too far apart. 3 goes into 84. 3 goes into 8. 2 times 3 is 6. 8 minus 6 is 2. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | 3 goes into 84. 3 goes into 8. 2 times 3 is 6. 8 minus 6 is 2. Bring down the 4. It goes exactly 8 times. So 3 and 28. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | 8 minus 6 is 2. Bring down the 4. It goes exactly 8 times. So 3 and 28. This seems interesting. 3 and 28. And remember, one of these has to be negative. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | So 3 and 28. This seems interesting. 3 and 28. And remember, one of these has to be negative. So if we have negative 3 plus 28, that is equal to 25. Now, we found our two numbers, but it's not going to be quite as simple of an operation as what we did when this wasn't a 1 or when this was a 1 or a negative 1. What we're going to do now is split up this term right here. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | And remember, one of these has to be negative. So if we have negative 3 plus 28, that is equal to 25. Now, we found our two numbers, but it's not going to be quite as simple of an operation as what we did when this wasn't a 1 or when this was a 1 or a negative 1. What we're going to do now is split up this term right here. We're going to split this up into negative... We're going to split it up into positive 28x minus 3x. We're just going to split that term. That term is that term right there. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | What we're going to do now is split up this term right here. We're going to split this up into negative... We're going to split it up into positive 28x minus 3x. We're just going to split that term. That term is that term right there. And of course, you have your minus 21 there. And you have your 4x squared over here. Now, you might say, how did you pick the 28 to go here and the negative 3 to go there? |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | That term is that term right there. And of course, you have your minus 21 there. And you have your 4x squared over here. Now, you might say, how did you pick the 28 to go here and the negative 3 to go there? And it actually does matter. The way I thought about it is 3 or negative 3 and 21 or negative 21, they have some common factors in particular. They have the factor 3 in common. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Now, you might say, how did you pick the 28 to go here and the negative 3 to go there? And it actually does matter. The way I thought about it is 3 or negative 3 and 21 or negative 21, they have some common factors in particular. They have the factor 3 in common. And 28 and 4 have some common factors. So I kind of grouped the 28 on the side of the 4. And you're going to see what I mean in a second. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | They have the factor 3 in common. And 28 and 4 have some common factors. So I kind of grouped the 28 on the side of the 4. And you're going to see what I mean in a second. If we literally group these, so that term becomes 4x squared plus 28x. And then this side over here in pink, well, I could say it's plus negative 3x minus 21. Once again, I picked these. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | And you're going to see what I mean in a second. If we literally group these, so that term becomes 4x squared plus 28x. And then this side over here in pink, well, I could say it's plus negative 3x minus 21. Once again, I picked these. I grouped the negative 3 with the 21 or the negative 21 because they're both divisible by 3. And I grouped the 28 with the 4 because they're both divisible by 4. And now, in each of these groups, we factor as much out as we can. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Once again, I picked these. I grouped the negative 3 with the 21 or the negative 21 because they're both divisible by 3. And I grouped the 28 with the 4 because they're both divisible by 4. And now, in each of these groups, we factor as much out as we can. So both of these terms are divisible by 4x. So this orange term is equal to 4x times x. 4x squared divided by 4x is just x. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | And now, in each of these groups, we factor as much out as we can. So both of these terms are divisible by 4x. So this orange term is equal to 4x times x. 4x squared divided by 4x is just x. Plus 28x divided by 4x is just 7. Now, this second term, remember, you factor out everything that you can factor out. Well, both of these terms are divisible by 3 or negative 3. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | 4x squared divided by 4x is just x. Plus 28x divided by 4x is just 7. Now, this second term, remember, you factor out everything that you can factor out. Well, both of these terms are divisible by 3 or negative 3. So let's factor out a negative 3. And this becomes x plus 7. And now, something might pop out at you. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Well, both of these terms are divisible by 3 or negative 3. So let's factor out a negative 3. And this becomes x plus 7. And now, something might pop out at you. We have x plus 7 times 4x plus x plus 7 times negative 3. So we can factor out an x plus 7. We can factor out an x plus 7. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | And now, something might pop out at you. We have x plus 7 times 4x plus x plus 7 times negative 3. So we can factor out an x plus 7. We can factor out an x plus 7. This might not be completely obvious. You're probably not used to factoring out an entire binomial, but you could view this as like a. Or if you have 4xa minus 3a, you would be able to factor out an a. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | We can factor out an x plus 7. This might not be completely obvious. You're probably not used to factoring out an entire binomial, but you could view this as like a. Or if you have 4xa minus 3a, you would be able to factor out an a. And I could just leave this as a minus sign. Let me delete this plus right here because it's just minus 3. It's just minus 3. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Or if you have 4xa minus 3a, you would be able to factor out an a. And I could just leave this as a minus sign. Let me delete this plus right here because it's just minus 3. It's just minus 3. Plus negative 3, same thing as minus 3. So what can we do here? We have x plus 7 times 4x. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | It's just minus 3. Plus negative 3, same thing as minus 3. So what can we do here? We have x plus 7 times 4x. We have an x plus 7 times negative 3. Let's factor out the x plus 7. We get x plus 7 times 4x minus 3. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | We have x plus 7 times 4x. We have an x plus 7 times negative 3. Let's factor out the x plus 7. We get x plus 7 times 4x minus 3. Minus that 3 right there. And we've factored our binomial. Sorry, we've factored our quadratic by grouping, and we factored it into two binomials. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | We get x plus 7 times 4x minus 3. Minus that 3 right there. And we've factored our binomial. Sorry, we've factored our quadratic by grouping, and we factored it into two binomials. Let's do another example of that and it's a little bit involved, but once you get the hang of it, it's kind of fun. So let's say we want to factor 6x squared plus 7x plus 1. Same drill. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Sorry, we've factored our quadratic by grouping, and we factored it into two binomials. Let's do another example of that and it's a little bit involved, but once you get the hang of it, it's kind of fun. So let's say we want to factor 6x squared plus 7x plus 1. Same drill. We want to find a times b that is equal to 1 times 6, and we want to find an a plus b needs to be equal to 7. This is a little bit more straightforward. The obvious one is 1 and 6. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Same drill. We want to find a times b that is equal to 1 times 6, and we want to find an a plus b needs to be equal to 7. This is a little bit more straightforward. The obvious one is 1 and 6. 1 times 6 is 6. 1 plus 6 is 7. So we have a is equal to 1. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | The obvious one is 1 and 6. 1 times 6 is 6. 1 plus 6 is 7. So we have a is equal to 1. Let me not even assign them. The numbers here are 1 and 6. Now we want to split this into a 1x and a 6x, but we want to group it so it's on the side of something that it shares a factor with. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | So we have a is equal to 1. Let me not even assign them. The numbers here are 1 and 6. Now we want to split this into a 1x and a 6x, but we want to group it so it's on the side of something that it shares a factor with. So we're going to have a 6x squared here plus, and so I'm going to put the 6x first because 6 and 6 share a factor. Then we're going to have plus 1x. 6x plus 1x is 7x. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Now we want to split this into a 1x and a 6x, but we want to group it so it's on the side of something that it shares a factor with. So we're going to have a 6x squared here plus, and so I'm going to put the 6x first because 6 and 6 share a factor. Then we're going to have plus 1x. 6x plus 1x is 7x. That was the whole point. They had to add up to 7. Then we have the final plus 1 there. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | 6x plus 1x is 7x. That was the whole point. They had to add up to 7. Then we have the final plus 1 there. Now in each of these groups, we can factor out as much as we like. So in this first group, let's factor out a 6x. So this first group becomes 6x times 6x squared divided by 6x is just an x. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Then we have the final plus 1 there. Now in each of these groups, we can factor out as much as we like. So in this first group, let's factor out a 6x. So this first group becomes 6x times 6x squared divided by 6x is just an x. 6x divided by 6x is just a 1. Then in the second group, we're going to have a plus here, but this second group, we just literally have an x plus 1, or we could even write a 1 times an x plus 1. You can imagine I just factored out a 1, so to speak. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | So this first group becomes 6x times 6x squared divided by 6x is just an x. 6x divided by 6x is just a 1. Then in the second group, we're going to have a plus here, but this second group, we just literally have an x plus 1, or we could even write a 1 times an x plus 1. You can imagine I just factored out a 1, so to speak. Now I have 6x times x plus 1 plus 1 times x plus 1. Well, I can factor out the x plus 1. If I factor out an x plus 1, that's equal to x plus 1 times 6x plus that 1. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | You can imagine I just factored out a 1, so to speak. Now I have 6x times x plus 1 plus 1 times x plus 1. Well, I can factor out the x plus 1. If I factor out an x plus 1, that's equal to x plus 1 times 6x plus that 1. I'm just doing the distributive property in reverse. So hopefully you didn't find that too bad. Now I'm going to actually explain why this little magical system actually works. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | If I factor out an x plus 1, that's equal to x plus 1 times 6x plus that 1. I'm just doing the distributive property in reverse. So hopefully you didn't find that too bad. Now I'm going to actually explain why this little magical system actually works. Why it actually works. Let me take an example. Let's say I have, well, I'll do it in very general terms. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Now I'm going to actually explain why this little magical system actually works. Why it actually works. Let me take an example. Let's say I have, well, I'll do it in very general terms. Let's add ax plus b times cx. Actually, I don't want to use, well, I'm afraid to use the a's and the b's. I think that'll confuse you because I use a's and b's here, and they won't be the same thing. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Let's say I have, well, I'll do it in very general terms. Let's add ax plus b times cx. Actually, I don't want to use, well, I'm afraid to use the a's and the b's. I think that'll confuse you because I use a's and b's here, and they won't be the same thing. Let me use completely different letters. Let's say I have fx plus g times hx plus, I'll use j instead of i. You'll learn in the future why I don't like using i as a variable. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | I think that'll confuse you because I use a's and b's here, and they won't be the same thing. Let me use completely different letters. Let's say I have fx plus g times hx plus, I'll use j instead of i. You'll learn in the future why I don't like using i as a variable. So what is this going to be equal to? Well, it's going to be fx times hx, which is fhx, and then fx times j, so plus fjx. Then we're going to have g times hx, so plus ghx, and then g times j, plus gj. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | You'll learn in the future why I don't like using i as a variable. So what is this going to be equal to? Well, it's going to be fx times hx, which is fhx, and then fx times j, so plus fjx. Then we're going to have g times hx, so plus ghx, and then g times j, plus gj. Or if we add these two middle terms, if you add the two middle terms, you have fh times x plus, add these two terms, fj plus ghx, plus gj. Now, what did I do here? Well, remember, in all of these problems where you have a non-1 or non-negative-1 coefficient, we look for two numbers that add up to this whose product is equal to the product of that times that. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Then we're going to have g times hx, so plus ghx, and then g times j, plus gj. Or if we add these two middle terms, if you add the two middle terms, you have fh times x plus, add these two terms, fj plus ghx, plus gj. Now, what did I do here? Well, remember, in all of these problems where you have a non-1 or non-negative-1 coefficient, we look for two numbers that add up to this whose product is equal to the product of that times that. Well, here we have two numbers that add up. Let's say that a is equal to fj. Let's say that a is equal to fj. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Well, remember, in all of these problems where you have a non-1 or non-negative-1 coefficient, we look for two numbers that add up to this whose product is equal to the product of that times that. Well, here we have two numbers that add up. Let's say that a is equal to fj. Let's say that a is equal to fj. That is a, and b is equal to gh. So a plus b is going to be equal to that middle coefficient. a plus b is going to be equal to that middle coefficient there. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Let's say that a is equal to fj. That is a, and b is equal to gh. So a plus b is going to be equal to that middle coefficient. a plus b is going to be equal to that middle coefficient there. And then what is a times b? a times b is going to be equal to fj times gh, which we could just reorder these terms. We're just multiplying a bunch of terms, so that could be rewritten as f times h times g times j. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | a plus b is going to be equal to that middle coefficient there. And then what is a times b? a times b is going to be equal to fj times gh, which we could just reorder these terms. We're just multiplying a bunch of terms, so that could be rewritten as f times h times g times j. These are all the same things. Well, what is fh times gj? This is equal to fh times gj. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | We're just multiplying a bunch of terms, so that could be rewritten as f times h times g times j. These are all the same things. Well, what is fh times gj? This is equal to fh times gj. Well, this is equal to the first coefficient times the constant term. So if a plus b will be equal to the middle coefficient, then a times b will equal the first coefficient times the constant term. So that's why this whole factoring by grouping even works, or how we're able to figure out what a and b even are. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | This is equal to fh times gj. Well, this is equal to the first coefficient times the constant term. So if a plus b will be equal to the middle coefficient, then a times b will equal the first coefficient times the constant term. So that's why this whole factoring by grouping even works, or how we're able to figure out what a and b even are. Now I'm going to close up with something slightly different, but just to make sure that you have a well-rounded education in factoring things. What I want to do is teach you to factor things a little bit more completely. This is a little bit of an add-on. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | So that's why this whole factoring by grouping even works, or how we're able to figure out what a and b even are. Now I'm going to close up with something slightly different, but just to make sure that you have a well-rounded education in factoring things. What I want to do is teach you to factor things a little bit more completely. This is a little bit of an add-on. I was going to make a whole video on this, but I think on some level it might be a little obvious for you. So let's say we had 2... Let me get a good one here. Let's say we had negative x to the third plus 17x squared minus 70. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | This is a little bit of an add-on. I was going to make a whole video on this, but I think on some level it might be a little obvious for you. So let's say we had 2... Let me get a good one here. Let's say we had negative x to the third plus 17x squared minus 70. Now, I have 70x. Now, immediately you say, gee, this isn't even a quadratic. I don't know how to solve something like this. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | Let's say we had negative x to the third plus 17x squared minus 70. Now, I have 70x. Now, immediately you say, gee, this isn't even a quadratic. I don't know how to solve something like this. It has an x to the third power. The first thing you should realize is that every term here is divisible by x. So let's factor out an x, or even better, let's factor out a negative x. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | I don't know how to solve something like this. It has an x to the third power. The first thing you should realize is that every term here is divisible by x. So let's factor out an x, or even better, let's factor out a negative x. So if you factor out a negative x, this is equal to negative x times... Negative x to the third divided by negative x is x squared. 17x squared divided by negative x is negative 17x. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | So let's factor out an x, or even better, let's factor out a negative x. So if you factor out a negative x, this is equal to negative x times... Negative x to the third divided by negative x is x squared. 17x squared divided by negative x is negative 17x. Negative 70x divided by negative x is positive 70. These cancel out. And now you have something that might look a little bit familiar. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | 17x squared divided by negative x is negative 17x. Negative 70x divided by negative x is positive 70. These cancel out. And now you have something that might look a little bit familiar. We have just a standard quadratic where the leading coefficient is a 1, so we just have to find 2 numbers whose product is 70 and that add up to negative 17. And the numbers that immediately jumped into my head are negative 10 and negative 7. You take their product, you get 70, you add them up, you get negative 17. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | And now you have something that might look a little bit familiar. We have just a standard quadratic where the leading coefficient is a 1, so we just have to find 2 numbers whose product is 70 and that add up to negative 17. And the numbers that immediately jumped into my head are negative 10 and negative 7. You take their product, you get 70, you add them up, you get negative 17. So this part right here is going to be x minus 10 times x minus 7. And of course, you have that leading negative x. The general idea here is just see if there's anything you can factor out and then it will get into a form that you might recognize. |
More examples of factoring by grouping Algebra I Khan Academy.mp3 | You take their product, you get 70, you add them up, you get negative 17. So this part right here is going to be x minus 10 times x minus 7. And of course, you have that leading negative x. The general idea here is just see if there's anything you can factor out and then it will get into a form that you might recognize. Hopefully you found this helpful. Now, I want to reiterate what I showed you at the beginning of this video. I think it's a really cool trick, so to speak, to be able to factor things that have a non-1 or non-negative 1 leading coefficient. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | And let's say that g of x, g of x, is equal to the cube root of x plus one, the cube root of x plus one minus seven. Now what I want to do now is evaluate f of, f of g of x, I want to evaluate f of g of x, and I also want to evaluate g of f of x, g of f of x, and see what I get. And I encourage you, like always, pause the video and try it out. All right, let's first evaluate f of g of x. So that means g of x, this expression, is going to be our input. So everywhere we see an x in the definition for f of x, we would replace it with all of g of x. So f of g of x is going to be equal to, so it's going to be equal to, well I see an x right over there, so I'd write all of g of x there. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | All right, let's first evaluate f of g of x. So that means g of x, this expression, is going to be our input. So everywhere we see an x in the definition for f of x, we would replace it with all of g of x. So f of g of x is going to be equal to, so it's going to be equal to, well I see an x right over there, so I'd write all of g of x there. So that's the cube root of x plus one minus seven, and then I have plus seven, plus seven, to the third power minus one. Notice, wherever I saw the x, since I'm taking f of g of x, I replace it with what g of x is. And so that is the cube root of x plus one minus seven. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | So f of g of x is going to be equal to, so it's going to be equal to, well I see an x right over there, so I'd write all of g of x there. So that's the cube root of x plus one minus seven, and then I have plus seven, plus seven, to the third power minus one. Notice, wherever I saw the x, since I'm taking f of g of x, I replace it with what g of x is. And so that is the cube root of x plus one minus seven. All right, now let's see if we can simplify this. Well we have a minus seven plus seven, so that simplifies nicely. So this just becomes, this is equal to, I can do it in a neutral color now, this is equal to the cube root of x plus one to the third power minus one. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | And so that is the cube root of x plus one minus seven. All right, now let's see if we can simplify this. Well we have a minus seven plus seven, so that simplifies nicely. So this just becomes, this is equal to, I can do it in a neutral color now, this is equal to the cube root of x plus one to the third power minus one. Well if I take the cube root of x plus one and then I raise it to the third power, well that's just going to give me x plus one. So this part, this part just simplifies to x plus one, and then I subtract one. So it all simplified out to just being equal to x. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | So this just becomes, this is equal to, I can do it in a neutral color now, this is equal to the cube root of x plus one to the third power minus one. Well if I take the cube root of x plus one and then I raise it to the third power, well that's just going to give me x plus one. So this part, this part just simplifies to x plus one, and then I subtract one. So it all simplified out to just being equal to x. So we're just left with an x. So f of g of x is just x. So now let's try what g of f of x is. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | So it all simplified out to just being equal to x. So we're just left with an x. So f of g of x is just x. So now let's try what g of f of x is. So g of f of x is going to be equal to, I'll do it right over here, this is going to be equal to the cube root of, and actually let me write it out. Wherever I see an x, I can write f of x instead. I didn't do it that last time. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | So now let's try what g of f of x is. So g of f of x is going to be equal to, I'll do it right over here, this is going to be equal to the cube root of, and actually let me write it out. Wherever I see an x, I can write f of x instead. I didn't do it that last time. I went directly and replaced with the definition of f of x. But just to make it clear what I'm doing. So everywhere I'm seeing an x, I replace it with an f of x. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | I didn't do it that last time. I went directly and replaced with the definition of f of x. But just to make it clear what I'm doing. So everywhere I'm seeing an x, I replace it with an f of x. So the cube root of f of x plus one minus seven. Well that's going to be equal to the cube root of, cube root of f of x, which is all of this business over here. So that is x plus seven to the third power minus one, and then we add one. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | So everywhere I'm seeing an x, I replace it with an f of x. So the cube root of f of x plus one minus seven. Well that's going to be equal to the cube root of, cube root of f of x, which is all of this business over here. So that is x plus seven to the third power minus one, and then we add one. And we add one, and then we subtract the seven. So lucky for us, this subtracting one and adding one, those cancel out. And so we're going to take the cube root of x plus seven to the third power. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | So that is x plus seven to the third power minus one, and then we add one. And we add one, and then we subtract the seven. So lucky for us, this subtracting one and adding one, those cancel out. And so we're going to take the cube root of x plus seven to the third power. Well the cube root of x plus seven to the third power is just going to be x plus seven. So this is going to be x plus seven. So all of this business simplifies to x plus seven. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | And so we're going to take the cube root of x plus seven to the third power. Well the cube root of x plus seven to the third power is just going to be x plus seven. So this is going to be x plus seven. So all of this business simplifies to x plus seven. And then we need to subtract seven. And these two cancel out, or they negate each other, and we are just left with x. So we see something very interesting. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | So all of this business simplifies to x plus seven. And then we need to subtract seven. And these two cancel out, or they negate each other, and we are just left with x. So we see something very interesting. f of g of x is just x, and g of f of x is x. So if we, in this case, if we start with an x, if we start with an x, we input it into the function g, and we get g of x, we get g of x, and then we input that into the function f, and we put that into the function f, f of g of x gets us back to x. It gets us back to x. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | So we see something very interesting. f of g of x is just x, and g of f of x is x. So if we, in this case, if we start with an x, if we start with an x, we input it into the function g, and we get g of x, we get g of x, and then we input that into the function f, and we put that into the function f, f of g of x gets us back to x. It gets us back to x. So we kind of did a round trip. And the same thing is happening over here. If I put x into f of x, if I put x into f of, sorry, if I put x into the function f, and I get f of x, the output is f of x, and then I input that into g, into the function g, into the function g, once again I do this round trip, and I get back to x. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | It gets us back to x. So we kind of did a round trip. And the same thing is happening over here. If I put x into f of x, if I put x into f of, sorry, if I put x into the function f, and I get f of x, the output is f of x, and then I input that into g, into the function g, into the function g, once again I do this round trip, and I get back to x. Another way to think about it, another way to think about it, if you view this as, so these are both composite functions, but one way to think about it is, if these are the set of all possible inputs into either of these composite functions, and then these are the outputs, so you are starting with an x, you are starting with an x, I'll do this case first. So g is a mapping, let me write down, so g is going to be a mapping from x to g of x. So this is what g is doing. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | If I put x into f of x, if I put x into f of, sorry, if I put x into the function f, and I get f of x, the output is f of x, and then I input that into g, into the function g, into the function g, once again I do this round trip, and I get back to x. Another way to think about it, another way to think about it, if you view this as, so these are both composite functions, but one way to think about it is, if these are the set of all possible inputs into either of these composite functions, and then these are the outputs, so you are starting with an x, you are starting with an x, I'll do this case first. So g is a mapping, let me write down, so g is going to be a mapping from x to g of x. So this is what g is doing. So the function g maps from x to some value g of x, g of x, and then if you were to apply f to this value right over here, if you apply f to this value, to g of x, you get all the way back to x. So that is f of g of x. And vice versa. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | So this is what g is doing. So the function g maps from x to some value g of x, g of x, and then if you were to apply f to this value right over here, if you apply f to this value, to g of x, you get all the way back to x. So that is f of g of x. And vice versa. If you start with x, and you apply f of x first, so if you start with f, if you apply f of x first, let me do that. So if you apply f of x first, you say you get to this value, so that is f of x. So you applied the function f, but then you apply the function g to that, you apply the function g to that, you get back. |
Verifying inverse functions by composition Mathematics III High School Math Khan Academy.mp3 | And vice versa. If you start with x, and you apply f of x first, so if you start with f, if you apply f of x first, let me do that. So if you apply f of x first, you say you get to this value, so that is f of x. So you applied the function f, but then you apply the function g to that, you apply the function g to that, you get back. So this is g of f of x, I should say, g of f, where we're applying the function g to the value f of x. And so since we get a round trip either way, we know that the functions g and f are inverses of each other. In fact, we can write, we can write that f of x is equal to the inverse of g of x, inverse of g of x, and vice versa. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | We need to factor negative 12 f squared minus 38f plus 22. So a good place to start is to see if, is there any common factor for all three of these terms? When you look at them, they're all even. And we don't like a negative number out here. So let's divide everything, or let's factor out a negative 2. So this expression right here is the same thing as negative 2 times negative 12 f squared divided by negative 2. So it's positive 6 f squared. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | And we don't like a negative number out here. So let's divide everything, or let's factor out a negative 2. So this expression right here is the same thing as negative 2 times negative 12 f squared divided by negative 2. So it's positive 6 f squared. Negative 38 divided by negative 2 is positive 19. So it'll be positive 19f. And then 22 divided by negative 2 is negative 11. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | So it's positive 6 f squared. Negative 38 divided by negative 2 is positive 19. So it'll be positive 19f. And then 22 divided by negative 2 is negative 11. So we've simplified it a bit. We have the 6 f squared plus 19f minus 11. We'll just focus on that part right now. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | And then 22 divided by negative 2 is negative 11. So we've simplified it a bit. We have the 6 f squared plus 19f minus 11. We'll just focus on that part right now. And the best way to factor this thing, since we don't have a 1 here as the coefficient on the f squared, is to factor it by grouping. So we need to look for two numbers whose products is 6 times negative 11. So two numbers. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | We'll just focus on that part right now. And the best way to factor this thing, since we don't have a 1 here as the coefficient on the f squared, is to factor it by grouping. So we need to look for two numbers whose products is 6 times negative 11. So two numbers. So a times b needs to be equal to 6 times negative 11, or negative 66. And a plus b needs to be equal to 19. So let's try a few numbers here. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | So two numbers. So a times b needs to be equal to 6 times negative 11, or negative 66. And a plus b needs to be equal to 19. So let's try a few numbers here. So let's see, 22, 22. I'm just thinking of numbers that are roughly 19 apart, because they're going to be of different signs. So 22 and 3, I think, will work. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | So let's try a few numbers here. So let's see, 22, 22. I'm just thinking of numbers that are roughly 19 apart, because they're going to be of different signs. So 22 and 3, I think, will work. Right. If we take 22 times negative 3, that is negative 66. And 22 plus negative 3 is equal to 19. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | So 22 and 3, I think, will work. Right. If we take 22 times negative 3, that is negative 66. And 22 plus negative 3 is equal to 19. And the way I kind of got pretty close to this number is, well, they're going to be of different signs. So the positive versions of them have to be about 19 apart. And that worked out. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | And 22 plus negative 3 is equal to 19. And the way I kind of got pretty close to this number is, well, they're going to be of different signs. So the positive versions of them have to be about 19 apart. And that worked out. So 22 and negative 3. So now we can rewrite this 19f right here as the sum of negative 3f and 22f. So it's negative 3f plus 22f. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | And that worked out. So 22 and negative 3. So now we can rewrite this 19f right here as the sum of negative 3f and 22f. So it's negative 3f plus 22f. That's the same thing as 19f. I just kind of broke it apart. And of course, we have the 6f squared. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | So it's negative 3f plus 22f. That's the same thing as 19f. I just kind of broke it apart. And of course, we have the 6f squared. And we have the minus 11 here. Now, you're probably saying, hey, Sal, why did you put the 22 here and the negative 3 there? Why didn't you do it the other way around? |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | And of course, we have the 6f squared. And we have the minus 11 here. Now, you're probably saying, hey, Sal, why did you put the 22 here and the negative 3 there? Why didn't you do it the other way around? Why didn't you put the 22 and then the negative 3 there? And my main motivation for doing it, I like to put the negative 3 on the same side with the 6, because they have the common factor of the 3. I like to put the 22 with the negative 11. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | Why didn't you do it the other way around? Why didn't you put the 22 and then the negative 3 there? And my main motivation for doing it, I like to put the negative 3 on the same side with the 6, because they have the common factor of the 3. I like to put the 22 with the negative 11. They have the same factor of 11. So that's why I decided to do it that way. So now let's do the grouping. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | I like to put the 22 with the negative 11. They have the same factor of 11. So that's why I decided to do it that way. So now let's do the grouping. And of course, you can't forget this negative 2 that we have sitting out here the whole time. So let me put that negative 2 out there. But that'll just kind of hang out for a while. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | So now let's do the grouping. And of course, you can't forget this negative 2 that we have sitting out here the whole time. So let me put that negative 2 out there. But that'll just kind of hang out for a while. But let's do some grouping. So let's group these first two. And then we're going to group this. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | But that'll just kind of hang out for a while. But let's do some grouping. So let's group these first two. And then we're going to group this. Let me get a nice color here. And then we're going to group this second 2. That's almost an identical color. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | And then we're going to group this. Let me get a nice color here. And then we're going to group this second 2. That's almost an identical color. Let me do it in this purple color. And then we can group that second 2 right there. So these first two, we could factor out a negative 3f. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | That's almost an identical color. Let me do it in this purple color. And then we can group that second 2 right there. So these first two, we could factor out a negative 3f. So it's negative 3f times 6f squared divided by negative 3f is negative 2f. And the negative 3f divided by negative 3f is just positive f. Actually, a better way to start, instead of factoring out a negative 3f, let's just factor out 3f. So we don't have a negative out here. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | So these first two, we could factor out a negative 3f. So it's negative 3f times 6f squared divided by negative 3f is negative 2f. And the negative 3f divided by negative 3f is just positive f. Actually, a better way to start, instead of factoring out a negative 3f, let's just factor out 3f. So we don't have a negative out here. We could do it either way. But if we just factor out a 3f, 6f squared divided by 3f is 2f. And then negative 3f divided by 3f is negative 1. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | So we don't have a negative out here. We could do it either way. But if we just factor out a 3f, 6f squared divided by 3f is 2f. And then negative 3f divided by 3f is negative 1. So that's what that factors into. And then that second part, in that dark purple color, we can factor out an 11. So we factor out an 11. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | And then negative 3f divided by 3f is negative 1. So that's what that factors into. And then that second part, in that dark purple color, we can factor out an 11. So we factor out an 11. And if we factor that out, 22f divided by 11 is 2f. And negative 11 divided by 11 is negative 1. And of course, once again, you have that negative 2 hanging out there. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | So we factor out an 11. And if we factor that out, 22f divided by 11 is 2f. And negative 11 divided by 11 is negative 1. And of course, once again, you have that negative 2 hanging out there. You have that negative 2. Now, inside the parentheses, we have two terms, both of which have 2f minus 1 as a factor. So we can factor that out. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | And of course, once again, you have that negative 2 hanging out there. You have that negative 2. Now, inside the parentheses, we have two terms, both of which have 2f minus 1 as a factor. So we can factor that out. This whole thing is just an exercise in doing the reverse distributive property, if you will. So let's factor that out. So you have 2f minus 1 times this 3f, times that 3f, and then times that plus 11. |
Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 | So we can factor that out. This whole thing is just an exercise in doing the reverse distributive property, if you will. So let's factor that out. So you have 2f minus 1 times this 3f, times that 3f, and then times that plus 11. Let me do that in the same shade of purple right over there. And you can distribute, if you like. 2f minus 1 times 3f will give you this term. |
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