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Current time:0:00Total duration:4:29

Dividing polynomials by x (no remainders)

Video transcript

what I'd like to do in this video is try to figure out what X to the 4th minus 2x to the 3rd plus 5x divided by X is equal to so pause this video and see if you can have a go at that before we work through this together all right so if we're saying what is this top expression divided by this bottom expression another way to think about it is what do I have to multiply so I'm going to multiply something I'll put that in parenthesis if I multiply that something times X I should get X to the 4th minus 2x to the 3rd plus 5x now how do I approach that well there's two ways that I could tackle it one way is I could just rewrite this expression as being e and I will just make this X in yellow so I can keep track of it I could just rewrite this as 1 over X x times X to the 4th minus 2x to the 3rd plus 5x and then I can distribute the 1 over X and so what is that going to be equal to what's going to be equal to X to the 4th let me do this X to the 4th over X minus 2 X to the 3rd over X plus 5x plus 5x over X and so what are each of these going to be equal to X to the fourth divided by X if I have four X's that I'm multiplying together and then I divide by X that's going to be equivalent to X to the third power so this right over here is equal to X to the third you could also get there from your exponent properties in the denominator you have an X to the first power and so you would subtract the exponents you have the same base here so that's X to the third and then in this part right over here what would that equal to well it's going to be minus 2x to the third divided by X to the first well by the same property that's going to be x squared and then last but not least if you take five X's and then you divide by X you are just going to be left with five and you can verify that this indeed if I were to multiply it by X I'm going to get X to the fourth minus 2 X to the third plus 5x let me do that if I put X to the third minus two x squared plus five times X what I can do is distribute the X X times X to the third is X to the fourth X times negative two x squared is negative two x to the third x times five is five x now I mentioned there's two ways that I could do it another way that I could try to tackle it is I could look at this numerator and try to factor an X out I would try to factor out whatever I see in the denominator so if I do that actually let me just rewrite the numerator so I can rewrite X to the fourth as x times X to the third and then I can rewrite the minus 2x to the third as let me write it this way as plus x times negative two x squared and then I could write this 5x as being equal to plus X times five and then I'm going to divide everything by X divide everything by X I just rewrote the numerator here but for each of those terms I factored out an X and now I can factor out X out of the whole thing so I sometimes think of factoring out an X out of the whole thing is reverse distributive property so if i factor out this x out of every term what am I left with I'm left with an X times X to the third minus two x squared plus five ended up doing that in the wrong in the same color but hopefully you're you're following plus five and then all of that is divided by X and as long as X does not equal zero X divided by X is going to be equal to one and we're left with what we had to begin with or the answer that we had to begin with so these are two different approaches nothing super super sophisticated here when you're dividing by X you're just okay that's the same thing as multiplying every term by one over X or you can factor out an X out of the numerator and then they cancel out