If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# 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? • 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. • At isnt the value for number of P(t)= 0 4 as their are 4 brackets so they could be 0?
(1 vote) • Wait, what if c is negative? Then would the first expression be correct or closer to correct?
(1 vote) • 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) • 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) 