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# Analyzing polynomial identities

CCSS.Math:

## Video transcript

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 are 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 say oh well you know it's obviously from this step to this step in you're going to try to follow it it's like well does that make sense so it's a really useful muscle to be able to see like does do these steps or how whoever manipulated the polynomial that doesn't 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 3 times X minus 2 squared and it looks like this person over five steps tries to expand it out and so what I encourage you to 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 for to step one what do they do well they just expand it out X minus 2 squared X minus 2 squared is just X minus 2 times X minus 2 they haven't done anything to the 4 X minus 3 yet and I want to do in a step so that seems correct so in step 2 they looks like they're just trying to multiply X minus 2 times X minus 2 so you have X times X which would be x squared you have x times negative 2 which would be negative 2x you have negative 2 times X which would be negative 2x and then you have negative 2 times negative 2 which would be positive 4 so it looks like they multiply this out correctly so step 2 words we're still doing good all right now what are they do in step 3 and this whole time for X minus 3 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 it's going to be negative 4x so this still this still looks correct the x squared didn't change the plus 4 didn't change they just added these middle two terms now as we go to this next for well now they're trying to multiply 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 just 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 going to be negative 16x squared so they did that right then you have 4x times 4 which is going to be 16x and they wrote that right over there then you're going to have negative 3 times x squared which is negative 3 x squared we see that right over there then you're going to have a negative 3 times negative 4x which is going to be positive 12x and they wrote negative 12x so they forgot they saw me they saw a negative negative but they still put a negative there negative 3 times negative 4x is negative x I thing is going to be a positive positive 12x so they made an error here and then they said negative 3 times positive 4 which is indeed negative 12 so this part is right so the error this thing should read positive positive 12x so the error they made is in step 4 step 4 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 the negative 12x plus 16x got you this 4x but we know it's supposed to be plus 12x so it really should be it really should be 28 this should be 20 this should be 28 X right over here but they really messed up if you if you take that error they did this step right but step 4 is where they actually made the error so let's keep going let's give ourselves a little bit more practice at at looking at ways to manipulate polynomials and see if they're valid so here this comes from an exercise on Khan Academy let's see which of these are valid identities which are which of these are valid statements so this first one 2x plus y times 4 X 2 X plus y times 4 X minus 2y is all of this business right over here let's just multiply it out 2x times 4x is going to be 8x squared 2x times negative 2 y is going to be negative negative 4 XY and then let me switch colors Y times 4x is going to be plus 4 X Y and then Y times negative 2 y is going to be is going to be minus 2 y squared and so let's see these two did I do that right let's see 2x 2x times negative 2 is negative 4 X Y and then I had 4 x times y is positive 4 X Y so these two are going to cancel out so this is already this is looking shady so while we're left with is 8 x squared minus 2y squared if we factor 2 out if we factor 2 out it's going to be 2 times 4 x squared minus y squared so this is this is not a true this is not a true statement right over here now let's try this one n plus 2 squared n plus 2 squared minus N squared is equal to this well what's n plus 2 squared that's going to be N squared plus plus 4n it's going to be 2 n plus 2 n it's going to be plus 4 n plus 4 and then we're going to subtract out an N squared these cancel so you're gonna have 4n plus 4 which is equal to 4 times n plus 1 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 1 is going to be plus a and then if I have B times 2a it's going to be plus 2 a B and then finally if I have B eight times one it's going to be plus B and then out here and then out here we are subtracting a B so over here we're going to subtract to be these characters are going to cancel out and then we're left with two a squared plus a plus two a B and it looks like they factored out an a so let's see if we can factor out an a ourselves so if we factor out an a we're going to be left with this first term it's going to be 2a it's 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 is going to be 2 B and that's what exactly what they wrote over here they just wrote in a different order a times 2a 2a plus 2b plus 2b plus 1 plus 1 so this is legit so hopefully that gave us some good practice it's 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