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Dividing rational expressions

Sal divides and simplifies (2p+6)/(p+5) ➗ (10)/(4p+20). Created by Sal Khan and Monterey Institute for Technology and Education.

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  • leaf green style avatar for user Nick Leoutsakos
    I am re-asking JPRandall's question since it is unanswered after after 7 months and I also want to understand this... can someone at least give a hint as to where to find a proof for this?
    his question:
    "Over the last few videos you say the domain of the simplified expression must be the same as the original expression, to make it mathematically equivalent. Although it makes sense to me, could you perhaps give a more rigorous reason, is it to do with the definition of a function for example?"
    (46 votes)
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    • leafers ultimate style avatar for user GFauxPas
      The definition of a function involves several things. The main thing is how the number inputted (the argument) is associated with the number outputted (the image). But part of the definition of a function is the domain. Very often the domain is only implied but part of the definition of a function is its domain. If you are changing the function to a form where the domain isn't implied that you have to state it explicitly. There's nothing to prove, it's just a definition.
      You can always state the domain, and it's important to do so when you're afraid it's not implied. For example, If you made a function involving money, you might have to say explicitly that the domain is all numbers such that the number of cents is an integer, so you don't have fractions of a cent.
      (34 votes)
  • blobby green style avatar for user TheGrayJedi
    Hey Sal when you get the answer at , it is acceptable to write the answer as 4p+12/5 or do we have to leave in the factor state?
    (15 votes)
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  • male robot donald style avatar for user Nick Temmel
    How come you don't change the signs of the variables and numbers when you flip them on the second fraction? And do you make Khan Academy for iPhone or iPod, because that would be great.
    and comment on this to tell me because you would be the best person ever!
    (6 votes)
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    • hopper cool style avatar for user Chuck Towle
      When you flip the variable, what you are really doing is using its inverse.
      When you multiply a number by its inverse, the result is always 1.

      For instance 1/2 * 2/1 = 1 so 2 is the inverse of 1/2.
      And -1/2 * -2/1 = 1, so -1/2 is the inverse of -2.

      The inverse of a negative number is always negative, becasue it takes two negative numbers multiplied togather to get a postive 1.
      And the inverse of a positive number is always positive,

      So when you flip a fraction to use its inverse, you would leave the sign the same.

      I hope that helps it click for you.
      (9 votes)
  • primosaur seedling style avatar for user john.doe.13896
    If you divide by a fraction, must always the numerator and denomiator be non-zero, since a / (b/c) = a * c/b?
    (3 votes)
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    • duskpin sapling style avatar for user Vu
      If a÷b/c then b≠0 c≠0

      If b=0 then a÷(0/c)=a/0 this is undefined.
      If c=0 then a÷(b/0), b/0 is undefined.
      Therefore if a÷b/c then b≠0 c≠0
      a÷b/c=a(c/b) for b≠0 and c≠0
      You must always add that restriction to make it true.

      However when you learn about limit of functions, you'll find that the polynomial of the denominator equals to zero. We then are allowed to rationalize it by algebraically manipulate it because we aren't finding the value that makes the denominator 0, but the value that approaches it. And also it's why we restrict the domain where the denominator is 0.
      (4 votes)
  • piceratops ultimate style avatar for user Batman
    What is the domain and why is it helpful?
    (3 votes)
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  • leafers seedling style avatar for user VAISHNAV
    I would like to get more practices on division of rational numbers. If anyone knows where I can get them pls reply
    (4 votes)
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  • male robot hal style avatar for user Marioland
    At why would you need to put 4p+20=0 if you are going to take the reciprocal? Wouldn't the 4p+20=0 be on top so you wouldn't have to worry about it being a zero?
    (4 votes)
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  • leafers seedling style avatar for user Abigail Conrad
    At when Sal reached the answer, 4(p+3) over 5, providing p does not equal -5, he did not further simplify the equation. Can you not simplify it into 4p+12 over 5, providing that p does not equal -5?
    (3 votes)
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    • purple pi purple style avatar for user doctorfoxphd
      In more advanced math, you often have to take your result and do other steps. That is usually easier if the result is in factored form. Sometimes in Calculus you will want to multiply the result all the way out. Practice will help you choose...and also little hints like the instructions you receive with the assignment and choices you are given as possible answers.
      (4 votes)
  • leaf green style avatar for user Lira Greene Isaura
    why didn't he do 2p + 6 can not = 0
    2p does not equal -6/2 =-3 so p can not equal -3 as well? :/
    (3 votes)
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  • blobby green style avatar for user JParkerRandall
    Over the last few videos you say the domain of the simplified expression must be the same as the original expression, to make it mathematically equivalent. Although it makes sense to me, could you perhaps give a more rigorous reason, is it to do with the definition of a function for example?
    (3 votes)
    Default Khan Academy avatar avatar for user

Video transcript

Divide and express as a simplified rational. State the domain. We start off with this expression. We actually have one rational expression divided by another rational expression. And like we've seen multiple times before, these rational expressions aren't defined when their denominators are equal to 0. So p plus 5 cannot be equal to 0, or if we subtract both sides of this-- we can't call it an equation, but we could call it a not-equation-- by negative 5, if we subtract negative 5 from both sides, you get p cannot be equal to-- these cancel out-- negative 5. That's what that tells us. Over here, we could do the same exercise 4p plus 20 also cannot be equal to 0. If it was, this expression would be undefined. Subtract 20 from both sides. 4p cannot be equal to negative 20. Divide both sides by 4. p cannot be equal to negative 5. So in both situations, p being equal to negative 5 would make either of these rational expressions undefined. So the domain here is the set of all reals such-- or p is equal to the set of all reals such that p does not equal negative 5, or essentially all numbers except for negative 5, all real numbers. We've stated the domain, so now let's actually simplify this expression. When you divide by a fraction or a rational expression, it's the same thing as multiplying by the inverse. Let me just rewrite this thing over here. 2p plus 6 over p plus 5 divided by 10 over 4p plus 20 is the same thing as multiplying by the reciprocal here, multiplying by 4p plus 20 over 10. I changed the division into a multiplication and I flipped this guy right here. Now, this is going to be equal to 2p plus 6 times 4p plus 20 in the numerator. I won't skip too many steps. Let me just write that. 2p plus 6 times 4p plus 20 in the numerator and then p plus 5 times 10 in the denominator. Now, in order to see if we can simplify this, we need to completely factor all of the terms in the numerator and the denominator. In the numerator, 2p plus 6, we can factor out a 2, so the 2p plus 6 we can rewrite it as 2 times p plus 3. Then the 4p plus 20, we can rewrite that. We can factor out a 4 as-- so 4 times p plus 5. Then we have our p plus 5 down there in the denominator. We have this p plus 5. We can just write it down in the denominator. Even 10, we can factor that further into its prime components or into its prime factorization. We can factor 10 into 2 times 5. That's the same thing as 10. Let's see what we can simplify. Of course, this whole time, we have to add the caveat that p cannot equal negative 5. We have to add this restriction on the domain in order for it to be the same expression as the one we started off with. Now, what can we cancel out? We have a 2 divided by a 2. Those cancel out. We have a p plus 5 divided by a p plus 5. We know that p plus 5 isn't going to be equal to 0 because of this constraint, so we can cancel those out. What are we left with? In the numerator, we have 4 times p plus 3, and in the denominator, all we have is that green 5, and we're done! We could right this as 4/5 times p plus 3, or just the way we did it right there. But we don't want to forget that we have to add the constraint p cannot be equal to negative 5, so that this thing is mathematically equivalent to this thing right here.