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Polynomial identities introduction

Video transcript

what we're going to do in this video is talk a little bit about polynomial identities and this is really just a fancy way of seeing whether an expression that involves a polynomial is equal to another expression so for example you're familiar with x squared plus 2x plus one we've seen polynomials like this multiple times this is a quadratic and you might recognize that this would be equal to X plus 1 squared that for any value of x x squared plus 2x plus 1 is the same thing as adding 1 to that X and then squaring the whole thing and we saw this when we first got when we first learned how to multiply binomials and and we saw and we took the square of binomials but now we're going to do this with slightly more complicated expressions things that aren't just simple quadratics or that might not be as obvious as this and the way that we're going to prove whether they're true or not is just put a little bit of algebraic manipulation so for example if someone walked up to you on the street and said alright M to the third minus 1 is it equal to M minus 1 times 1 plus M plus M Squared pause this video and see what you would tell that person whether you could prove whether it is or is not a true polynomial identity okay let's do it together and the way I would tackle this is I would expand out I would multiply out what we have on the right hand side so this is going to be equal to so first I could take this M and then multiply it times every term in this second expression so M times 1 is M M times M is M Squared and then M times M Squared is M to the third power and then I would take this negative 1 and then multiply and then distribute that times every term in that other expression so negative 1 times 1 is negative 1 negative 1 times M is negative M and negative 1 times M Squared is negative M Squared and now let's see if we can simplify this we have an M and a negative M so those are going to cancel out we have an M Squared and an negative M Squared so those cancel out and so we are going to be left with em to the third power minus-1 now clearly em to the third power minus one is going to be equal to n to the third power minus one for any value of M these are these are identical expressions so this is this is indeed a polynomial identity let's do another example let's say someone were to walk up to you on the street and said quick n plus three squared plus two n is that equal to eight n plus 13 is this a polynomial identity pause this video and see if you could figure that out alright now we're gonna work on that together and I would do it the exact same way I would try to simplify with a little bit of algebra the maybe the easiest thing to do first and you could do this in multiple ways is I have this I have these n terms to ends here eight ends over here well what if I were to get these two ends out of the left hand side so if I were to just subtract two n from both sides of this equation I am going to get on the left hand side n plus three squared and on the right hand side I am going to get six n8 n minus 2n plus 13 now what's n plus three squared well that's going to be N squared plus two times 3 times n if what I just did it does not seem familiar to you I encourage you to look at the videos about squaring binomials but this is going to be plus 6 n plus 3 squared which is 9 and is this going to be equal to 6 n plus 13 well already this is starting to look a little bit a little bit sketchy but let's just keep going with it with the algebra so let's see if we subtract 6 n from both sides what do you get well on the left hand side you're going to have N squared plus 9 and on the right hand side you're going to get 13 now are there values of n for which this is not always true well sure I can find a lot of values of n for which this is not always true and is a 0 this is not going to be true if n is 1 this is not going to be true if n is 2 this actually would be true but if n is 3 this is not going to be true if n is 4 or 5 etcetera so for actually most values of n this is not going to be true so in order for it to be a polynomial identity it has to be true for all of the values that are legitimate values that you can evaluate for those for the variable in question so this run right over here is not a polynomial polynomial identity and we're done