Proving polynomial identities
- [Voiceover] What I hope to do in this video is give ourselves some practice at critically looking at how folks manipulate polynomials. And the reason why this is useful is because it's useful to be able to do this to yourself as you manipulate polynomials to say wait, what did I exactly do there? Or a lot of times if you're reading a math or a science book they're going to do some proof or something like that, and they're gonna say oh well you know, it's obviously from this step to this step, and you're gonna try to follow it and you're just like, well does that make sense? So it's a really useful muscle to be able to see do these steps, or whoever manipulated the polynomial, does it make sense to you? And especially if it's you, it's super useful to be able to find if there are errors, and to correct them, it'll just give you a better critical eye for this type of thing. So let's just start with this one. We have 4x minus three times x minus two squared. And it looks like this person over five steps tries to expand it out. And so what I encourage you to do, pause the video right now, and see if they did it correctly. And if they didn't do it correctly, try to identify on what step they messed up. All right, so assuming you had a go at it, let's do this together. So as we go from the first expression to the second, to step one, what do they do? Well, they just expanded out x minus two squared. X minus two squared is just x minus two times x minus two, they haven't done anything to the 4x minus three yet. Now what do they do in this step, so that seems correct. So in step two, it looks like they're just trying to multiply x minus two times x minus two. So you have x times x, which would be x squared. You have x times negative two, which would be negative 2x. You have negative two times x, which would be negative 2x. Then you have negative two times negative two, which would be positive four. So it looks like they multiplied this out correctly, so step two we're still doing good. All right, now what do they do in step three? And this whole time, 4x minus three, they haven't really touched it yet. So they're just trying to simplify it, and all they did is they added these two middle terms. Minus negative 2x minus 2x is going to be negative 4x, so this still, this still looks correct. The x squared didn't change, the plus four didn't change, they just added these middle two terms. Now as we go to this next step four, well now they're trying to multiply these two expressions, so they're doing some algebraic multiplication. So let's see if we can figure this out. So we have 4x times, let me do this in a new color, I'm getting bored of that magenta. All right, so we have, we have 4x times x squared, which is indeed 4x to the third power. Then you have 4x times negative 4x, which is gonna be negative 16x squared, so they did that right. Then you have 4x times four, which is going to be 16x, and they wrote that right over there. Then, you're going to have negative three times x squared, which is negative 3x squared, we see that right over there. Then you're going to have negative three times negative 4x, which is going to be positive 12x, and they wrote negative 12x. So they forgot, they saw a negative, negative, but they still put a negative there. Negative three times negative 4x, a negative times a negative is gonna be a positive. Positive 12x. So they made an error here, and then they said negative three times positive four, which is indeed negative 12, so this part is right. So the error, this thing should read positive 12x. So the error they made is in step four. Step four is the error, and then that ended up giving them the wrong answer here, because they did a minus 12x instead of, if this was a negative 12x, then negative 12x plus 16x got you this 4x. But we know it's supposed to be plus 12x. So it really should be 28. This should be 28x right over here. But they really messed up. If you take that error, they did this step right. But step four is where they actually made the error. So let's keep going, let's give ourselves a little bit more practice at looking at ways to manipulate polynomials and see if they're valid. So here, this comes from an exercise on Kahn Academy. Let's see which of these are valid identities. Which of these are valid statements? So this first one, 2x plus y, times 4x, 2x plus y times 4x minus 2y is all of this business right over here. Le'ts just multiply it out. 2x times 4x is going to be 8x squared. 2x times negative 2y is going to be negative 4xy. And then, let me switch colors, y times 4x is going to be plus 4xy. And then y times negative 2y is going to be minus 2y squared. And so let's see, these two, did I do that right? Let's see, 2x, times negative 2y is negative 4xy. And then I had 4x times y is positive 4xy. So these two are gonna cancel out. So this is already, this is looking shady. So all we're left with is 8x squared minus 2y squared, if we factor a two out, it's going to be two times 4x squared minus y squared. So this is not a true statement right over here. Now let's try this one. N plus two squared minus n squared is equal to this. But what's n plus two squared? That's going to be n squared plus plus 4n, it's gonna be 2n plus 2n, it's gonna be plus 4n plus four, and then we're gonna subtract out an n squared, these cancel, so we're gonna have 4n plus four, which is equal to four times n plus one. So this one right over here works out. This is a true, I guess in the language of this question, it is a valid identity, or you could say it's a true statement, this equation is true. And then we have this last one right over here. And once again, let's see if we can multiply it out. Let me go down here into the black space. So if I have a times 2a, that's going to be 2a squared. And then a times one, is going to be plus a. And then if I have b times 2a, it's going to be plus 2ab, and then finally if I have b times one, it's going to be plus b. And then out here, we are subtracting a b. So over here we're going to subtract a b. These characters are going to cancel out. And then we're left with 2a squared plus a plus 2ab. And it looks like they factor out an a, so let's see if we can factor out an a ourselves. So if we factor out an a, we're gonna be left with this first term is going to be 2a, this right over here, if we factor out an a, is going to be plus one, and this if we factor out an a is going to be 2b. And that's exactly what they wrote over here, they just wrote it in a different order. A times 2a, plus 2b plus one. So this is legit. So hopefully that gave us some good practice at critically looking at whether people are making valid statements. And this is going to be the most useful to figure out if you are making valid statements. All right, hopefully you enjoyed that.