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# Structure in rational expression

CCSS.Math:

## Video transcript

let's say what we're told that the position of a particle can be figured out from or though the position the position of a particle is a function of time is given by this crazy expression the position could be positive or negative and this expression is T minus a times a minus T times T minus a times t minus B all of that over the square root of a squared plus B squared and they also tell us that C is greater than B is greater than a is greater than zero so given that information we have two statements right over here we have the position at time C on the left hand side and over here we have the number of times that our function is equal to zero the number of times that P of T is equal to zero what I want you to do is pause the video right now and think about which of these statements provides a larger value is P of C greater than or less than than the number of times P of T equals zero and I guess a third and fourth option would be that you maybe don't have enough information to figure this out or maybe that these are equal so I encourage you to figure that out right now which of these are larger or do you not have enough information or are they equal so I'm assuming you've got given a go at it so let's think about each of them so let's think about what P of C is so P of C is going to be equal to right here P I don't want to arbitrarily switch colors which I sometimes do P of C P of C is going to be equal to let's see it's going to be C minus a I'll just solve this one color it's going to be C minus a times a minus C times C minus a times C minus B all of that over all that over the square root of a squared plus B squared so what do we know about this quantity what do we know about this quantity right over here let me highlight all the C's here so C minus a a minus C C minus AC minus B well they tell us that C is larger than a and B and that they're all positive so maybe we can come up with some statement about whether this thing is positive or negative whether what whether this expression is so what's C minus a going to be well C is greater than a so this is going to be positive what about a minus C well a is less than C so this is going to result in a negative number C minus a well this is going to be a positive again and then C minus B is also going to be a positive C is greater than both B and a and what do we have here in the denominator well the square root of a squared plus B squared well that's this is just going to be this is going to be a positive value so what do we have going on here here in the numerator I have a positive times a negative times a positive times a positive so what's that going to be well that's going to be a negative a positive times a positive times a positive is positive and then you throw that negative in there so you're going to get you're going to have a negative over a positive and what's a negative divided by a positive well that's going to be a negative negative so we don't know what this actual value is but all we do know is that this provides us with a negative negative value so if this ends up being a positive value then we can make a statement if this we just say hey this is a negative value as well that we might not have enough information so let's think about the number of times P of T is equal to zero well P of T is equal to zero whenever the numerator right over here is equal to zero and when would the numerator equals zero well have the product I have the product of this one two three four four expression so if any one of these expressions is zero then the entire numerator is going to be zero so let's think about how you can make these expressions zero so the expression could be 0 this will be 0 if T is equal to a this would also be 0 of T is equal to a and this would also be 0 of T is equal to a and this would be 0 if T is equal to B so there's two values that would make T 2 values for T that will make this numerator equal 0 T equals a or T equals B so there's two times so the number of times that P of T is equal to 0 is is 2 so now let's answer our question what is larger the number two the number positive two versus some some negative number well 2 is larger than any negative number so this this is the larger quantity