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## Precalculus

### Unit 4: Lesson 5

Modeling with rational functions- Analyzing structure word problem: pet store (1 of 2)
- Analyzing structure word problem: pet store (2 of 2)
- Combining mixtures example
- Rational equations word problem: combined rates
- Rational equations word problem: combined rates (example 2)
- Mixtures and combined rates word problems
- Rational equations word problem: eliminating solutions
- Reasoning about unknown variables
- Reasoning about unknown variables: divisibility
- Structure in rational expression

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# Structure in rational expression

CCSS.Math: , , ,

Analyzing an elaborate rational expression to determine what's bigger: the value of that expression at some unknown c, or the number of times the expression is equal to 0? This is algebraic reasoning at its best! Created by Sal Khan.

## Want to join the conversation?

- If p(c) is negative, therefore less than 0, why is it neccesary to ascertain the number of times that p(t)=0 when there can't be negative times p(t)=0 and therefore even if p(t) never equals zero, it will still be more than a negative number?(54 votes)
- Sherlock, you are right: there is no need for us to determine the exact number of times when p(t)=0 to answer the question. I believe Sal worked through the second part of the problem just to show us the intuition and to have some fun:)(34 votes)

- I'm not seeing the big picture. It doesn't seem like the solution here tells us anything important about the problem itself.(5 votes)
- The point here is that by looking at the problem in a way where we are concerned with only certain characteristics of the variables (in this case, which of the terms in the numerator end up positive, and which end up negative), we can get some information about how the function behaves over certain parts of the domain of the function. Knowing that when using an input value that is larger than the variable a and the variable b in the function, the result is always going to be negative, might be a powerful bit of information, depending on what the problem is. In this example, where the function is a position function, I think it is telling us that when we take the position at a time that is bigger than a or b, the position has moved backwards, in the negative direction. We can also get information about which direction the position goes when using a time value smaller than the variable a, or between the variable a and b. I can't see the original problem as I'm writing this, but I think the numerator was (t-a)(a-t)(t-a)(t - b) with the constraint that b>a>0. So if we pick a c that is smaller than a, the 1st term is negative, the 2nd term is positive, the third term is negative, and the fourth term is negative, so overall the numerator is negative. But if we pick a value between a and b. The first term is positive, the second term is negative, the third term is positive, and the fourth term is negative, so the numerator is positive.(26 votes)

- Isn't it that the square root of (a^2+b^2) is a positive and a negative number?(8 votes)
- That's the principal square root you saw there in the denominator. So it is always positive. You would be right if you saw the plus/minus sign before the square root. A single minus sign would also make it negative.(6 votes)

- Why does this seem difficult and unsolvable when Sal asks to pause the video and tackle it ourselves, but then it's easily understood when he starts explaining it?! What is it of the mindset that I may be missing?(4 votes)
- Practice, practice, practice - for a lot of people, the more you work on math, the more you learn number sense, inductive and deductive reasoning, and more intuitive approaches, so you will be able to do more and more as you go. When you first are learning a new concept, unless it is closely related to something you know, then it will be hard to solve, but I could probably give you a lot of 4th and 5th grade problems that you could work quickly without much deep thinking. Enjoy Math for all of its patterns and logic, and you will be able to start tackling it yourself.(8 votes)

- I understand the concept and how to solve the problem, but how can this aid me in solving actual problems with rational expressions?(3 votes)
- You need to explain how c can be negative. It clearly states c > b > a > 0 This is just fanciful maths there is no logic to it! A number less than zero is negative not a number greater than 0.(2 votes)
- c
**is not**negative, but the function p, evaluated at c, that is, p(c), gives a negative number. This is fine.

I suggest you try the problem with some numbers, eg,

let a=1, b=2 and c=3, for which c > b > a > 0, that is, 3 > 2 > 1 > 0, correct?

Now evaluate p(3) and see what happens.(2 votes)

- At4:30isnt the value for number of P(t)= 0 4 as their are 4 brackets so they could be 0?(1 vote)
- While there are 4 brackets, there are only 2 distinct possibilities because (c - a)( a - c) and (c-a) are all the same c = a.(2 votes)

- Wait, what if c is negative? Then would the first expression be correct or closer to correct?(1 vote)
- C cannot be negative because in the second statement it says that it is greater than 0.(2 votes)

- how o we know the sqrt of a^2+b^2 is positive? Two the sqrt of a positive can be either positive or negative(1 vote)
- no, anything squared will be positive. Say for ex.
`-2^2`

which equals`4`

. Because if a negative multiplies a negative it will be positive, same for positives. So if a positive adds to a positive it will be positive. Hope it helps.(1 vote)

- Couldn't we have used deductive reasoning to state that as the number of values that would make p(t) equal to 0 can't be negative (this value has a minimum of 0) this value is greater than c without even finding the number of values that would make p(t) equal to 0?(1 vote)

## 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.