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# Structure in rational expression

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?
• 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:)
• I'm not seeing the big picture. It doesn't seem like the solution here tells us anything important about the problem itself.
• 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.
• Isn't it that the square root of (a^2+b^2) is a positive and a negative number?
• 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.
• 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?
• 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.
• I understand the concept and how to solve the problem, but how can this aid me in solving actual problems with rational expressions?
• 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.
• 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.
• At isnt 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.
• This Question took me forever to answer :
Malik walked up and back down an empty ‍18m long escalator at a constant rate. Since the escalator was going up at the time, it took him only ‍22.5s Going down took Malik‍ 90s Assume the escalator speed is constant.

What was Malik's walking speed?

This is the link to the problem and their solution : https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:rational-functions/x9e81a4f98389efdf:modeling-with-rational-functions/e/mixtures-and-combined-rates-word-problems

How come they come up with some long winded solution when after thinking about it, all I had to do to get it right was set up an equation : 18m/22.5s + 18m/90s = malik's walking speed = 90m/180s = 0.5 m/s
I had been struggling on these problems till I finally busted down and got out my notebook and pencil. For most of home school I've been trying to do everything in my head.
So why does my solution work?
(1 vote)
• The solution they show allows us to see how it all works. Your intuition in setting up the equation as you have it is great, but with and understanding of why you will be a math and problem solving champ!
Using variables, we can see how this works:
d/M+E = 22.5
d/M-E = 90
Where d is distance,
M is Maliks speed,
E is escalator speed.
Rate is defined as displacement/time
which is what you have 18m/22.5
this is going to be equal to M+E
and 18/90 = M-E
if we were to substitute this we get:
M+E + M-E = 18/22.5 + 18/90
Simplify the left side:
2M = 18/22.5 + 18/90
simplify the right side:
2M = 1
M = 0.5
• 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.