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Plotting complex numbers on the complex plane Precalculus Khan Academy.mp3 | So we have a complex number here. It has a real part, negative 2. It has an imaginary part. You have 2 times i. And what you see here is we're going to plot it on this kind of two-dimensional grid, but it's not our traditional coordinate axes. In our traditional coordinate axis, you're plotting a real x value versus a real y coordinate. Here, on the horizontal axis, that's going to be the real part of our complex number, and our vertical axis is going to be the imaginary part. |
Plotting complex numbers on the complex plane Precalculus Khan Academy.mp3 | You have 2 times i. And what you see here is we're going to plot it on this kind of two-dimensional grid, but it's not our traditional coordinate axes. In our traditional coordinate axis, you're plotting a real x value versus a real y coordinate. Here, on the horizontal axis, that's going to be the real part of our complex number, and our vertical axis is going to be the imaginary part. So in this example, this complex number, our real part is the negative 2, and then our imaginary part is a positive 2. And so that right over there in the complex plane is the point negative 2 plus 2i. Let's do a few more of these. |
Plotting complex numbers on the complex plane Precalculus Khan Academy.mp3 | Here, on the horizontal axis, that's going to be the real part of our complex number, and our vertical axis is going to be the imaginary part. So in this example, this complex number, our real part is the negative 2, and then our imaginary part is a positive 2. And so that right over there in the complex plane is the point negative 2 plus 2i. Let's do a few more of these. So 5 plus 2i, once again, real part is 5, imaginary part is 2, and we're done. Let's do two more of these. 1 plus 5i. |
Plotting complex numbers on the complex plane Precalculus Khan Academy.mp3 | Let's do a few more of these. So 5 plus 2i, once again, real part is 5, imaginary part is 2, and we're done. Let's do two more of these. 1 plus 5i. 1, that's the real part, plus 5i right over that im. All right, let's do one more of these. 4 minus 4i. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | In this video, we're going to learn to divide polynomials. And sometimes this is called algebraic long division. But you'll see what I'm talking about when we do a few examples. Let's say I just want to divide 2x plus 4 and divide it by 2. And we're not really changing the value. We're just changing how we're going to express the value. So we already know how to simplify this. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | Let's say I just want to divide 2x plus 4 and divide it by 2. And we're not really changing the value. We're just changing how we're going to express the value. So we already know how to simplify this. We've done this in the past. We could divide the numerator and the denominator by 2. And this would be equal to what? |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | So we already know how to simplify this. We've done this in the past. We could divide the numerator and the denominator by 2. And this would be equal to what? This would be equal to x plus 2. Let me write it this way. It would be equal to, if you divide this by 2, it becomes an x. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | And this would be equal to what? This would be equal to x plus 2. Let me write it this way. It would be equal to, if you divide this by 2, it becomes an x. You divide the 4 by 2, it becomes a 2. If you divide the 2 by 2, you get a 1. So this is equal to x plus 2, which is pretty straightforward, I think. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | It would be equal to, if you divide this by 2, it becomes an x. You divide the 4 by 2, it becomes a 2. If you divide the 2 by 2, you get a 1. So this is equal to x plus 2, which is pretty straightforward, I think. The other way is you could have factored a 2 out of here, and then those would have canceled out. But I'll also show you how to do it using algebraic long division, which is a bit of overkill for this problem. But I just want to show you that it's not fundamentally anything new. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | So this is equal to x plus 2, which is pretty straightforward, I think. The other way is you could have factored a 2 out of here, and then those would have canceled out. But I'll also show you how to do it using algebraic long division, which is a bit of overkill for this problem. But I just want to show you that it's not fundamentally anything new. It's just a different way of doing things. But it's useful for more complicated problems. So you could have also written this as 2 goes into 2x plus 4 how many times? |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | But I just want to show you that it's not fundamentally anything new. It's just a different way of doing things. But it's useful for more complicated problems. So you could have also written this as 2 goes into 2x plus 4 how many times? And you would perform this the same way you would do traditional long division. You'd say 2, you'd always start with the highest degree term. 2 goes into the highest degree term. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | So you could have also written this as 2 goes into 2x plus 4 how many times? And you would perform this the same way you would do traditional long division. You'd say 2, you'd always start with the highest degree term. 2 goes into the highest degree term. You would ignore the 4. 2 goes into 2x how many times? Well, it goes into 2x x times. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | 2 goes into the highest degree term. You would ignore the 4. 2 goes into 2x how many times? Well, it goes into 2x x times. And you put the x in the x place. x times 2 is 2x. And just like traditional long division, you now subtract. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | Well, it goes into 2x x times. And you put the x in the x place. x times 2 is 2x. And just like traditional long division, you now subtract. You now subtract. So 2x plus 4 minus 2x is what? It's 4, right? |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | And just like traditional long division, you now subtract. You now subtract. So 2x plus 4 minus 2x is what? It's 4, right? And then 2 goes into 4 how many times? It goes into 2 times, a positive 2 times. Put that in the constants place. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | It's 4, right? And then 2 goes into 4 how many times? It goes into 2 times, a positive 2 times. Put that in the constants place. 2 times 2 is 4. You subtract. Remainder is 0. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | Put that in the constants place. 2 times 2 is 4. You subtract. Remainder is 0. So this might seem overkill for what was probably a problem that you already knew how to do and do it in a few steps. But we're now going to see that this is a very generalizable process. You can do this really for any degree polynomial, dividing into any other degree polynomial. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | Remainder is 0. So this might seem overkill for what was probably a problem that you already knew how to do and do it in a few steps. But we're now going to see that this is a very generalizable process. You can do this really for any degree polynomial, dividing into any other degree polynomial. Let me show you what I'm talking about. So let's say we wanted to divide x plus 1 into x squared plus 3x plus 6. So what do we do here? |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | You can do this really for any degree polynomial, dividing into any other degree polynomial. Let me show you what I'm talking about. So let's say we wanted to divide x plus 1 into x squared plus 3x plus 6. So what do we do here? So you look at the highest degree term here, which is an x. And you look at the highest degree term here, which is an x squared. So you can ignore everything else. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | So what do we do here? So you look at the highest degree term here, which is an x. And you look at the highest degree term here, which is an x squared. So you can ignore everything else. And that really simplifies the process. And you say x goes into x squared how many times? Well, x squared divided by x is just x, right? |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | So you can ignore everything else. And that really simplifies the process. And you say x goes into x squared how many times? Well, x squared divided by x is just x, right? x goes into x squared x times. And you put it in the x place. This is the x place right here, or the x to the first power place. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | Well, x squared divided by x is just x, right? x goes into x squared x times. And you put it in the x place. This is the x place right here, or the x to the first power place. So x times x plus 1 is what? x times x is x squared. x times 1 is x. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | This is the x place right here, or the x to the first power place. So x times x plus 1 is what? x times x is x squared. x times 1 is x. So it's x squared plus x. And just like we did over here, we now subtract. And what do we get? |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | x times 1 is x. So it's x squared plus x. And just like we did over here, we now subtract. And what do we get? x squared plus 3x plus 6 minus x squared. Let me be very careful. This is minus x squared plus x. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | And what do we get? x squared plus 3x plus 6 minus x squared. Let me be very careful. This is minus x squared plus x. Don't want to make sure that negative sign only applies to this whole thing. So x squared minus x squared, those cancel out. 3x, this is going to be a minus x. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | This is minus x squared plus x. Don't want to make sure that negative sign only applies to this whole thing. So x squared minus x squared, those cancel out. 3x, this is going to be a minus x. Let me put that sign there. So this is minus x squared minus x, just to be clear. We're subtracting the whole thing. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | 3x, this is going to be a minus x. Let me put that sign there. So this is minus x squared minus x, just to be clear. We're subtracting the whole thing. 3x minus x is 2x. And then you bring down the 6, or 6 minus 0 is nothing. So 2x plus 6. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | We're subtracting the whole thing. 3x minus x is 2x. And then you bring down the 6, or 6 minus 0 is nothing. So 2x plus 6. Now you look at the highest degree term, an x and a 2x. How many times does x go into 2x? It goes into it two times. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | So 2x plus 6. Now you look at the highest degree term, an x and a 2x. How many times does x go into 2x? It goes into it two times. 2 times x is 2x. 2 times 1 is 2. 2 times 1 is 2. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | It goes into it two times. 2 times x is 2x. 2 times 1 is 2. 2 times 1 is 2. So we get 2 times x plus 1 is 2x plus 2. But we're going to want to subtract this from this up here, so we're going to subtract it. Instead of writing 2x plus 2, we could just write negative 2x minus 2 and then add them. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | 2 times 1 is 2. So we get 2 times x plus 1 is 2x plus 2. But we're going to want to subtract this from this up here, so we're going to subtract it. Instead of writing 2x plus 2, we could just write negative 2x minus 2 and then add them. These guys cancel out. 6 minus 2 is 4. And how many times does x go into 4? |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | Instead of writing 2x plus 2, we could just write negative 2x minus 2 and then add them. These guys cancel out. 6 minus 2 is 4. And how many times does x go into 4? Well, we could just say that 0 times, or we could say that 4 is the remainder. So if we wanted to rewrite x squared plus 3x plus 6 over x plus 1, notice this is the same thing as x squared plus 3x plus 6 divided by x plus 1. This thing divided by this. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | And how many times does x go into 4? Well, we could just say that 0 times, or we could say that 4 is the remainder. So if we wanted to rewrite x squared plus 3x plus 6 over x plus 1, notice this is the same thing as x squared plus 3x plus 6 divided by x plus 1. This thing divided by this. We can now say that this is equal to x plus 2. It is equal to x plus 2 plus the remainder divided by x plus 1 plus 4 over x plus 1. This right here and this right here are equivalent. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | This thing divided by this. We can now say that this is equal to x plus 2. It is equal to x plus 2 plus the remainder divided by x plus 1 plus 4 over x plus 1. This right here and this right here are equivalent. And if you wanted to check that, if you wanted to go from this back to that, what you could do is multiply this by x plus 1 over x plus 1 and then add the 2. So this is the same thing as x plus 2. I'm just going to multiply that times x plus 1 over x plus 1, that's just multiplying it by 1. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | This right here and this right here are equivalent. And if you wanted to check that, if you wanted to go from this back to that, what you could do is multiply this by x plus 1 over x plus 1 and then add the 2. So this is the same thing as x plus 2. I'm just going to multiply that times x plus 1 over x plus 1, that's just multiplying it by 1. And then to that, add 4 over x plus 1. I did that so I have the same common denominator. And when you perform this addition right here, when you multiply these two binomials and then add the 4 up here, you should get x squared plus 3x plus 6. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | I'm just going to multiply that times x plus 1 over x plus 1, that's just multiplying it by 1. And then to that, add 4 over x plus 1. I did that so I have the same common denominator. And when you perform this addition right here, when you multiply these two binomials and then add the 4 up here, you should get x squared plus 3x plus 6. Let's do another one of these. They're kind of fun. So let's say that we want to simplify x squared plus 5x plus 4 over x plus 4. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | And when you perform this addition right here, when you multiply these two binomials and then add the 4 up here, you should get x squared plus 3x plus 6. Let's do another one of these. They're kind of fun. So let's say that we want to simplify x squared plus 5x plus 4 over x plus 4. So once again, we can do our algebraic long division. We can divide x plus 4 into x squared plus 5x plus 4. And once again, same exact process. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | So let's say that we want to simplify x squared plus 5x plus 4 over x plus 4. So once again, we can do our algebraic long division. We can divide x plus 4 into x squared plus 5x plus 4. And once again, same exact process. Look at the highest degree terms in both of them. x goes into x squared how many times? It goes into it x times. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | And once again, same exact process. Look at the highest degree terms in both of them. x goes into x squared how many times? It goes into it x times. Put in the x place. This is our x place right here. x times x is x squared. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | It goes into it x times. Put in the x place. This is our x place right here. x times x is x squared. x times 4 is 4x. And then of course, we're going to want to subtract these from there. So let me just put a negative sign there. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | x times x is x squared. x times 4 is 4x. And then of course, we're going to want to subtract these from there. So let me just put a negative sign there. And then these cancel out. 5x minus 4x is x. 4 minus 0 is plus 4. x plus 4, and you could even see this coming. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | So let me just put a negative sign there. And then these cancel out. 5x minus 4x is x. 4 minus 0 is plus 4. x plus 4, and you could even see this coming. You could say x plus 4 goes into x plus 4 obviously one time. Or you could just look at, if you were not looking at the constant terms, you would completely just say, well, x goes into x how many times? Well, one time. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | 4 minus 0 is plus 4. x plus 4, and you could even see this coming. You could say x plus 4 goes into x plus 4 obviously one time. Or you could just look at, if you were not looking at the constant terms, you would completely just say, well, x goes into x how many times? Well, one time. Plus 1. 1 times x is x. 1 times 4 is 4. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | Well, one time. Plus 1. 1 times x is x. 1 times 4 is 4. We're going to subtract them from up here. So it cancels out. So we have no remainder. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | 1 times 4 is 4. We're going to subtract them from up here. So it cancels out. So we have no remainder. So this right here simplifies to, this is equal to x plus 1. And there's other ways you could have done this. We could have tried to factor this numerator. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | So we have no remainder. So this right here simplifies to, this is equal to x plus 1. And there's other ways you could have done this. We could have tried to factor this numerator. x squared plus 5x plus 4 over x plus 4. This is the same thing as what? We could have factored this numerator as x plus 4 times x plus 1. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | We could have tried to factor this numerator. x squared plus 5x plus 4 over x plus 4. This is the same thing as what? We could have factored this numerator as x plus 4 times x plus 1. 4 times 1 is 4. 4 plus 1 is 5. All of that over x plus 4. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | We could have factored this numerator as x plus 4 times x plus 1. 4 times 1 is 4. 4 plus 1 is 5. All of that over x plus 4. That cancels out, and you're left just with x plus 1. Either way would have worked. But the algebraic long division will always work, even if you can't cancel out factors like that. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | All of that over x plus 4. That cancels out, and you're left just with x plus 1. Either way would have worked. But the algebraic long division will always work, even if you can't cancel out factors like that. Even if you did have remainder in this situation, you didn't. So this was equal to x plus 1. Let's do another one of these. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | But the algebraic long division will always work, even if you can't cancel out factors like that. Even if you did have remainder in this situation, you didn't. So this was equal to x plus 1. Let's do another one of these. Just to make sure that you really have, because this is actually a very, very useful skill to have in your toolkit. So let's say we have x squared. Let me just change it up. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | Let's do another one of these. Just to make sure that you really have, because this is actually a very, very useful skill to have in your toolkit. So let's say we have x squared. Let me just change it up. Let's say we had 2x squared. I could really make these numbers up on the fly. 2x squared minus 20x plus 12 divided by. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | Let me just change it up. Let's say we had 2x squared. I could really make these numbers up on the fly. 2x squared minus 20x plus 12 divided by. Actually, let's make it really interesting. Just to show you that it'll always work. I want to go above quadratic. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | 2x squared minus 20x plus 12 divided by. Actually, let's make it really interesting. Just to show you that it'll always work. I want to go above quadratic. So let's say we have 3x to the third minus 2x squared plus 7x minus 4. And we want to divide that. We want to divide that by x squared plus 1. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | I want to go above quadratic. So let's say we have 3x to the third minus 2x squared plus 7x minus 4. And we want to divide that. We want to divide that by x squared plus 1. I just made this up. But we can just do the algebraic long division to figure out what this is going to be, or what this is. We simplify it, what it'll be. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | We want to divide that by x squared plus 1. I just made this up. But we can just do the algebraic long division to figure out what this is going to be, or what this is. We simplify it, what it'll be. x squared plus 1 divided into this thing up here. 3x to the third minus 2x squared plus 7x minus 4. Once again, look at the highest degree term. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | We simplify it, what it'll be. x squared plus 1 divided into this thing up here. 3x to the third minus 2x squared plus 7x minus 4. Once again, look at the highest degree term. x squared goes into 3x to the third how many times? Well, it's going to go into it 3x times. You multiply 3x times this, you get 3x to the third. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | Once again, look at the highest degree term. x squared goes into 3x to the third how many times? Well, it's going to go into it 3x times. You multiply 3x times this, you get 3x to the third. So it's going to go into it 3x times. So you have to write the 3x over here in the x term. So it's going to go into it 3x times, just like that. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | You multiply 3x times this, you get 3x to the third. So it's going to go into it 3x times. So you have to write the 3x over here in the x term. So it's going to go into it 3x times, just like that. Now let's multiply. 3x times x squared is 3x to the third plus 3x times 1. So we have a 3x over here. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | So it's going to go into it 3x times, just like that. Now let's multiply. 3x times x squared is 3x to the third plus 3x times 1. So we have a 3x over here. I'm making sure to put it in the x place. And we're going to want to subtract them. And what do we have when we do that? |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | So we have a 3x over here. I'm making sure to put it in the x place. And we're going to want to subtract them. And what do we have when we do that? These cancel out. We have a minus 2x squared. And then 7x minus 3x is plus 4x. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | And what do we have when we do that? These cancel out. We have a minus 2x squared. And then 7x minus 3x is plus 4x. And we have a minus 4. Once again, look at the highest degree term. x squared and a negative 2x squared. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | And then 7x minus 3x is plus 4x. And we have a minus 4. Once again, look at the highest degree term. x squared and a negative 2x squared. So x squared goes into negative 2x squared negative 2 times. Put it in the constants place. Negative 2 times x squared is negative 2x squared. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | x squared and a negative 2x squared. So x squared goes into negative 2x squared negative 2 times. Put it in the constants place. Negative 2 times x squared is negative 2x squared. Negative 2 times 1 is negative 2. Now we're going to want to subtract these from there. So let's multiply them by negative 1, or those become a positive. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | Negative 2 times x squared is negative 2x squared. Negative 2 times 1 is negative 2. Now we're going to want to subtract these from there. So let's multiply them by negative 1, or those become a positive. These two guys cancel out. 4x minus 0 is 4x minus. Let me switch colors. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | So let's multiply them by negative 1, or those become a positive. These two guys cancel out. 4x minus 0 is 4x minus. Let me switch colors. 4x minus 0 is 4x. Negative 4 minus negative 2, or negative 4 plus 2, is equal to negative 2. And then x squared now has a higher degree than 4x, the highest degree here. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | Let me switch colors. 4x minus 0 is 4x. Negative 4 minus negative 2, or negative 4 plus 2, is equal to negative 2. And then x squared now has a higher degree than 4x, the highest degree here. So we view this as the remainder. So this expression we could rewrite it as being equal to 3x minus 2. That's the 3x minus 2. |
Polynomial division Polynomial and rational functions Algebra II Khan Academy.mp3 | And then x squared now has a higher degree than 4x, the highest degree here. So we view this as the remainder. So this expression we could rewrite it as being equal to 3x minus 2. That's the 3x minus 2. Plus our remainder, 4x minus 2. All of that over x squared plus 1. x squared plus 1. Hopefully you found that as fun as I did. |
Negative fractional exponent examples Algebra I Khan Academy.mp3 | And we know that because three times three is equal to nine. This is equivalent to saying, what is the principal root of nine? Well, that is equal to three. But what would happen if I took nine to the negative 1 1⁄2 power? Now we have a negative fractional exponent. And the key to this is to just not to get too worried or intimidated by this, but just think about it step by step. Just ignore for the second that this is a fraction. |
Negative fractional exponent examples Algebra I Khan Academy.mp3 | But what would happen if I took nine to the negative 1 1⁄2 power? Now we have a negative fractional exponent. And the key to this is to just not to get too worried or intimidated by this, but just think about it step by step. Just ignore for the second that this is a fraction. And just look at this negative first. Just breathe slowly and realize, OK, I got a negative. Negative exponent. |
Negative fractional exponent examples Algebra I Khan Academy.mp3 | Just ignore for the second that this is a fraction. And just look at this negative first. Just breathe slowly and realize, OK, I got a negative. Negative exponent. That means that this is just going to be 1 over 9 to the 1 1⁄2. That's what that negative is the cue for. This is 1 over 9 to the 1 1⁄2. |
Negative fractional exponent examples Algebra I Khan Academy.mp3 | Negative exponent. That means that this is just going to be 1 over 9 to the 1 1⁄2. That's what that negative is the cue for. This is 1 over 9 to the 1 1⁄2. And we know that 9 to the 1 1⁄2 is equal to 3. So this is just going to be equal to 1. This is just going to be equal to 1 3rd. |
Negative fractional exponent examples Algebra I Khan Academy.mp3 | This is 1 over 9 to the 1 1⁄2. And we know that 9 to the 1 1⁄2 is equal to 3. So this is just going to be equal to 1. This is just going to be equal to 1 3rd. Let's take things a little bit further. What would this evaluate to? And I encourage you to pause the video after trying it. |
Negative fractional exponent examples Algebra I Khan Academy.mp3 | This is just going to be equal to 1 3rd. Let's take things a little bit further. What would this evaluate to? And I encourage you to pause the video after trying it. Or pause the video to try it. Negative 27 to the negative 1 3rd power. So I encourage you to pause the video and think about what this would evaluate to. |
Negative fractional exponent examples Algebra I Khan Academy.mp3 | And I encourage you to pause the video after trying it. Or pause the video to try it. Negative 27 to the negative 1 3rd power. So I encourage you to pause the video and think about what this would evaluate to. So remember, just take a deep breath. You can always get rid of this negative in the exponent by taking the reciprocal and raising it to the positive. So this is going to be equal to 1 over negative 27 to the positive 1 3rd power. |
Negative fractional exponent examples Algebra I Khan Academy.mp3 | So I encourage you to pause the video and think about what this would evaluate to. So remember, just take a deep breath. You can always get rid of this negative in the exponent by taking the reciprocal and raising it to the positive. So this is going to be equal to 1 over negative 27 to the positive 1 3rd power. And I know what you're saying. Hey, I still can't breathe easily. I have this negative number to this fractional exponent. |
Negative fractional exponent examples Algebra I Khan Academy.mp3 | So this is going to be equal to 1 over negative 27 to the positive 1 3rd power. And I know what you're saying. Hey, I still can't breathe easily. I have this negative number to this fractional exponent. But this is just saying, what number, if I were to multiply it three times, so if I have that number, so whatever the number this is, I were to multiply it, if I took three of them and I multiply them together, if I multiplied 1 by that number three times, what number would I have to use here to get negative 27? Well, we already know that 3 to the 3rd, which is equal to 3 times 3 times 3 is equal to positive 27. So that's a pretty good clue. |
Negative fractional exponent examples Algebra I Khan Academy.mp3 | I have this negative number to this fractional exponent. But this is just saying, what number, if I were to multiply it three times, so if I have that number, so whatever the number this is, I were to multiply it, if I took three of them and I multiply them together, if I multiplied 1 by that number three times, what number would I have to use here to get negative 27? Well, we already know that 3 to the 3rd, which is equal to 3 times 3 times 3 is equal to positive 27. So that's a pretty good clue. What would negative 3 to the 3rd power be? Well, that's negative 3 times negative 3 times negative 3, which is negative 3 times negative 3 is positive 9 times negative 3 is negative 27. So we just found this number, this question mark. |
Negative fractional exponent examples Algebra I Khan Academy.mp3 | So that's a pretty good clue. What would negative 3 to the 3rd power be? Well, that's negative 3 times negative 3 times negative 3, which is negative 3 times negative 3 is positive 9 times negative 3 is negative 27. So we just found this number, this question mark. Negative 3 times negative 3 times negative 3 is equal to negative 27. So negative 27 to the 1 3rd, this part right over here, is equal to negative 3. So this is going to be equal to 1 over negative 3, which is the same thing as negative 1 3rd. |
Example Graphing y=3⋅sin(½⋅x)-2 Trigonometry Algebra 2 Khan Academy.mp3 | And it first bears mentioning how this widget works. So this point right over here, it helps you define the midline, the thing that you could imagine your sine or cosine function oscillates around. And then you also define a neighboring extreme point, either a maximum or a minimum point, to graph your function. So let's think about how we would do this. And like always, I encourage you to pause this video and think about how you would do it yourself. But the first way I like to think about it is what would a regular just... If this just said y is equal to sine of x, how would I graph that? |
Example Graphing y=3⋅sin(½⋅x)-2 Trigonometry Algebra 2 Khan Academy.mp3 | So let's think about how we would do this. And like always, I encourage you to pause this video and think about how you would do it yourself. But the first way I like to think about it is what would a regular just... If this just said y is equal to sine of x, how would I graph that? Well, sine of 0 is 0. Sine of pi over 2 is 1. And then sine of pi is 0 again. |
Example Graphing y=3⋅sin(½⋅x)-2 Trigonometry Algebra 2 Khan Academy.mp3 | If this just said y is equal to sine of x, how would I graph that? Well, sine of 0 is 0. Sine of pi over 2 is 1. And then sine of pi is 0 again. And so this is what just regular sine of x would look like. But let's think about how this is different. Well, first of all, it's not just sine of x. |
Example Graphing y=3⋅sin(½⋅x)-2 Trigonometry Algebra 2 Khan Academy.mp3 | And then sine of pi is 0 again. And so this is what just regular sine of x would look like. But let's think about how this is different. Well, first of all, it's not just sine of x. It's sine of 1 half x. So what would be the graph of just sine of 1 half x? Well, one way to think about it, there's actually two ways to think about it, is a coefficient right over here on your x term that tells you how fast the thing that's being inputted into sine is growing. |
Example Graphing y=3⋅sin(½⋅x)-2 Trigonometry Algebra 2 Khan Academy.mp3 | Well, first of all, it's not just sine of x. It's sine of 1 half x. So what would be the graph of just sine of 1 half x? Well, one way to think about it, there's actually two ways to think about it, is a coefficient right over here on your x term that tells you how fast the thing that's being inputted into sine is growing. And now it's going to grow half as fast. And so one way to think about it is your period is now going to be twice as long. So one way to think about it is instead of getting to this next maximum point at pi over 2, you're going to get there at pi. |
Example Graphing y=3⋅sin(½⋅x)-2 Trigonometry Algebra 2 Khan Academy.mp3 | Well, one way to think about it, there's actually two ways to think about it, is a coefficient right over here on your x term that tells you how fast the thing that's being inputted into sine is growing. And now it's going to grow half as fast. And so one way to think about it is your period is now going to be twice as long. So one way to think about it is instead of getting to this next maximum point at pi over 2, you're going to get there at pi. And you could test that. If you...when x is equal to pi, this will be 1 half pi. Sine of 1 half pi is indeed equal to 1. |
Example Graphing y=3⋅sin(½⋅x)-2 Trigonometry Algebra 2 Khan Academy.mp3 | So one way to think about it is instead of getting to this next maximum point at pi over 2, you're going to get there at pi. And you could test that. If you...when x is equal to pi, this will be 1 half pi. Sine of 1 half pi is indeed equal to 1. Another way to think about it is you might be familiar with the formula, although I always like you to think about where these formulas come from, that to figure out the period of a sine or cosine function, you take 2 pi and you divide it by whatever this coefficient is. So 2 pi divided by 1 half is going to be 4 pi. And you can see the period here, we go up, down, and back to where we were over 4 pi. |
Example Graphing y=3⋅sin(½⋅x)-2 Trigonometry Algebra 2 Khan Academy.mp3 | Sine of 1 half pi is indeed equal to 1. Another way to think about it is you might be familiar with the formula, although I always like you to think about where these formulas come from, that to figure out the period of a sine or cosine function, you take 2 pi and you divide it by whatever this coefficient is. So 2 pi divided by 1 half is going to be 4 pi. And you can see the period here, we go up, down, and back to where we were over 4 pi. And that makes sense because if you just had a 1 coefficient here, your period would be 2 pi. 2 pi radians, you make one circle around the unit circle is one way to think about it. So right here we have the graph of sine of 1 half x. |
Example Graphing y=3⋅sin(½⋅x)-2 Trigonometry Algebra 2 Khan Academy.mp3 | And you can see the period here, we go up, down, and back to where we were over 4 pi. And that makes sense because if you just had a 1 coefficient here, your period would be 2 pi. 2 pi radians, you make one circle around the unit circle is one way to think about it. So right here we have the graph of sine of 1 half x. Now what if we wanted to instead think about 3 times the graph of sine of 1 half x, or 3 sine 1 half x? Well, then our amplitude is just going to be 3 times as much. And so instead of our maximum point going from... instead of our maximum point being at 1, it will now be at 3. |
Example Graphing y=3⋅sin(½⋅x)-2 Trigonometry Algebra 2 Khan Academy.mp3 | So right here we have the graph of sine of 1 half x. Now what if we wanted to instead think about 3 times the graph of sine of 1 half x, or 3 sine 1 half x? Well, then our amplitude is just going to be 3 times as much. And so instead of our maximum point going from... instead of our maximum point being at 1, it will now be at 3. Or another way to think about it is we're going 3 above the midline and 3 below the midline. So this right over here is the graph of 3 sine of 1 half x. Now we have one thing left to do, and this is this minus 2. |
Example Graphing y=3⋅sin(½⋅x)-2 Trigonometry Algebra 2 Khan Academy.mp3 | And so instead of our maximum point going from... instead of our maximum point being at 1, it will now be at 3. Or another way to think about it is we're going 3 above the midline and 3 below the midline. So this right over here is the graph of 3 sine of 1 half x. Now we have one thing left to do, and this is this minus 2. So this minus 2 is just going to shift everything down by 2. So we just have to shift everything down. So let me shift this one down by 2, and let me shift this one down by 2. |
Example Graphing y=3⋅sin(½⋅x)-2 Trigonometry Algebra 2 Khan Academy.mp3 | Now we have one thing left to do, and this is this minus 2. So this minus 2 is just going to shift everything down by 2. So we just have to shift everything down. So let me shift this one down by 2, and let me shift this one down by 2. And so there you have it. Notice our period is still 4 pi. Our amplitude, how much we oscillate above or below the midline, is still 3. |
Example Graphing y=3⋅sin(½⋅x)-2 Trigonometry Algebra 2 Khan Academy.mp3 | So let me shift this one down by 2, and let me shift this one down by 2. And so there you have it. Notice our period is still 4 pi. Our amplitude, how much we oscillate above or below the midline, is still 3. And now we have this minus 2. Another way to think about it, when x is equal to 0, this whole first term is going to be 0, and y should be equal to negative 2. And we're done. |
Example Solving for a variable Linear equations Algebra I Khan Academy.mp3 | That looks like a rectangle. And if this side's length is L, then this side's length is also going to be L. And if this width is W, then this width up here is W. And the perimeter is just how, what is the distance if you were to go around this rectangle. And so that distance is going to be this W plus this L plus this W, or that width, plus this length. And if you have one W and you add it to another W, that's going to give you two Ws. So that's two Ws. And then if you have one L and then you have another L, that's going to give you, if you add them together, that's going to give you two Ls. So the perimeter is going to be two Ls plus two Ws. |
Example Solving for a variable Linear equations Algebra I Khan Academy.mp3 | And if you have one W and you add it to another W, that's going to give you two Ws. So that's two Ws. And then if you have one L and then you have another L, that's going to give you, if you add them together, that's going to give you two Ls. So the perimeter is going to be two Ls plus two Ws. They just wrote it in a different order than the way I wrote it. But the same thing. So hopefully that makes sense. |
Example Solving for a variable Linear equations Algebra I Khan Academy.mp3 | So the perimeter is going to be two Ls plus two Ws. They just wrote it in a different order than the way I wrote it. But the same thing. So hopefully that makes sense. Now their question is, rewrite the formula so that it solves for width. So the formula, the way it's written now, it says P is equal to something. They want us to write it, so it's this W right here, they want it to be W is equal to a bunch of stuff with Ls and Ps in it and maybe some numbers there. |
Example Solving for a variable Linear equations Algebra I Khan Academy.mp3 | So hopefully that makes sense. Now their question is, rewrite the formula so that it solves for width. So the formula, the way it's written now, it says P is equal to something. They want us to write it, so it's this W right here, they want it to be W is equal to a bunch of stuff with Ls and Ps in it and maybe some numbers there. So let's think about how we can do this. So they tell us that P is equal to 2 times L plus 2 times W. We want to solve for W. Well, a good starting point might be to get rid of the L on this side of the equation. And to get rid of it on that side of the equation, we could subtract the 2L from both sides of the equation. |
Example Solving for a variable Linear equations Algebra I Khan Academy.mp3 | They want us to write it, so it's this W right here, they want it to be W is equal to a bunch of stuff with Ls and Ps in it and maybe some numbers there. So let's think about how we can do this. So they tell us that P is equal to 2 times L plus 2 times W. We want to solve for W. Well, a good starting point might be to get rid of the L on this side of the equation. And to get rid of it on that side of the equation, we could subtract the 2L from both sides of the equation. So let's do it this way. So you subtract 2L over here, minus 2L. You're also going to have to do that on the left-hand side. |
Example Solving for a variable Linear equations Algebra I Khan Academy.mp3 | And to get rid of it on that side of the equation, we could subtract the 2L from both sides of the equation. So let's do it this way. So you subtract 2L over here, minus 2L. You're also going to have to do that on the left-hand side. So you're going to have minus 2L. We're doing it on both sides of the equation. Remember, an equation says P is equal to that. |
Example Solving for a variable Linear equations Algebra I Khan Academy.mp3 | You're also going to have to do that on the left-hand side. So you're going to have minus 2L. We're doing it on both sides of the equation. Remember, an equation says P is equal to that. So if you do anything to that, you have to do it to P. So if you subtract 2L from this, you're going to have to subtract 2L from P in order for the equality to keep being true. So the left-hand side is going to be P minus 2L. And then that is going to be equal to 2L minus 2L. |
Example Solving for a variable Linear equations Algebra I Khan Academy.mp3 | Remember, an equation says P is equal to that. So if you do anything to that, you have to do it to P. So if you subtract 2L from this, you're going to have to subtract 2L from P in order for the equality to keep being true. So the left-hand side is going to be P minus 2L. And then that is going to be equal to 2L minus 2L. The whole reason why we subtract the 2L is because these are going to cancel out. So these cancel out, and you're just left with a 2W here. We're almost there. |
Example Solving for a variable Linear equations Algebra I Khan Academy.mp3 | And then that is going to be equal to 2L minus 2L. The whole reason why we subtract the 2L is because these are going to cancel out. So these cancel out, and you're just left with a 2W here. We're almost there. We've almost solved for W. To finish it up, we just have to divide both sides of this equation by 2. And the whole reason why I'm dividing both sides of this equation by 2 is to get rid of this 2 coefficient. This 2 that's multiplying W. So if you divide both sides of this equation by 2, once again, if you do something to one side of the equation, you do it to the other side. |