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0 energy points

# Structure in rational expression

Video transcript

Let's say we we're told that
the position of a particle can be figured out
from, or the position of a particle as
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 0. So given that information,
we have two statements right over here. We have the position at times
c on the left hand side. And over here, we have
the number of times that our function is equal to 0,
the number of times that p of t is equal to 0. 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 the number of times p of t equals 0? 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
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 is going to be
equal to, let's see. It's going to be c minus a. I'll do this all
in 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 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 Cs here. So c minus a, a minus
c, c minus a, c 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 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, this is just
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 a
negative over a positive. And what's a negative
divided by a positive? Well, that's going
to be a negative. So we don't know what
the actual value is. But all we do know is
that this provides us with a negative value. So this ends up being
a positive value. Then we could make a statement
that this, we just say, hey, this is a negative
value as well. But we might not have
enough information. So let's think about the number
of times p of t is equal to 0. Well, p of t is
equal to 0 whenever the numerator right
over here is equal to 0. And when would the
numerator equal 0? Well, I have the product of
this 1, 2, 3, 4 expression. So if any one of these
expressions is 0, then the entire numerator
is going to be 0. So let's think about how you
can make these expressions 0. So the expression could be 0. This will be 0 if
t is equal to a. This would also be 0
if t is equal to a. And this would also be
0 if t is equal to a. And this would be 0
if t is equal to b. So there's two values
for t that will make this numerator equal to
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 two. So now let's answer
our question. What is larger, the number
2, is the number positive 2, versus some negative number? Well, 2 is larger than
any negative number. So this is the larger quantity.