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# Divide polynomials by x (with remainders)

CCSS.Math:

## Video transcript

simplify the expression 18x to the fourth minus 3x squared plus 6x minus 4 all of that over is 6x so there's a couple of ways to think about them they're all really equivalent you can really just view this up here as being the exact same thing as 18x to the fourth over 6x plus negative 3x squared over 6x or you could say minus 3x squared over 6x plus 6x over 6x minus 4 over 6x now there's a couple of ways to think about it one is I just kind of decomposed this numerator up here if I just had a bunch of stuff a plus B plus C over D that's clearly equal to a over D plus B over D plus C over D or maybe not so clearly but hopefully that helps clarify another way to think about it is kind of like you're distributing the division if I'm if I if I divide a whole expression by something that's equivalent to dividing each of the terms by that something the other way to think about it the other way to think about it is that we're multiplying this entire expression so this is the same thing as 1 over 6x times this entire thing times 18x to the 4th minus 3x squared plus 6x minus 4 and so here this would just be the straight distributive property to get to this whatever seems to make sense for you they're all equivalent they're all logical good things to do to simplify this thing now once you have it here now we just have a bunch of monomials that we're just dividing by 6x and here we can just use the exponent properties this first one over here we can take the coefficients and divide them 18 divided by 6 is 3 and then you have X to the fourth divided by X to the well they don't tell us but if we just an X that's the same thing as X to the first power so it's X to the fourth divided by X to the first that's going to be X to the 4 minus 1 power or X to the third power then we have this coefficient over here or these coefficients we have negative 3 divided by 6 so I'm going to do this part next negative 3 divided by six is negative 1/2 negative 1/2 and then you have x squared divided by X we already know that X is the same thing as X to the first so that's going to be X to the two minus one power which is just 1 or I could just leave it as an X right there then we have then we have these coefficients 6 divided by 6 well that's just 1 so I could just what I'll write it I could write a 1 here let me just write the 1 here because we said 2 minus 1 is 1 and then X divided by X is X to the first over X to the first you give you in some two ways anything divided by anything is just 1 or you could view it as X to the 1 divided by X to the 1 it's going to be X to the 1 minus 1 which is X to the 0 which is also equal to 1 either way you know how to do this before you even learn that exponent property because X divided by X is 1 and then assuming X does not equal 0 and then finally and we kind of have to assume X doesn't equal 0 and this whole thing otherwise we would be dividing by 0 and then finally we have 4 over 6 X and there's a couple of ways to think about it so well the simplest way is both 4 for negative 4 let me do it negative 4 over 6 is the same thing as negative 2/3 just simplify that fraction and we're multiplying that times 1 over X and so we could cut we could multiply that times 1 over X so we could view this 4 times 1 over X another way to think about it is you could have viewed this 4 is being multiplied by X to the 0 power and this being X to the first power and then when you try to simplify it using your exponent properties you would have well that would be X to the 0 minus 1 power which is X to the negative 1 power so we could have written an X to the negative 1 here but X to the negative 1 is the exact same thing as 1 over X and so let's just write our answer completely simplify it so it's going to be 3 X to the third minus 1/2 X plus 1 because this thing right here is just 1 so plus plus 1 and then minus 2 times 1 in the numerator over 3 times in the denominator and we are done or we could write this depending on what you consider more simplified this last term right here could also be written in minus two thirds x to the negative one but if you don't want a negative exponent you could write it like that