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Subtracting rational expressions: factored denominators

Sal subtracts two rational expressions whose denominators are factored. The denominators aren't the same but they share a factor.

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  • purple pi purple style avatar for user Mia Downs
    can you factor by grouping in the numerator at the end?
    (6 votes)
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  • blobby green style avatar for user Angie Ledon
    how do I stop getting confused between LCM and GFC
    (5 votes)
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  • duskpin tree style avatar for user harrypotter101_ella
    sorry i figured out that letter that i thought was a 2/7 is accually a z
    (6 votes)
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  • male robot hal style avatar for user Saavan Kiran
    Shouldn’t you check to see if the final answer can be simplified? Like maybe he could factor by grouping or something like that on the numerator and check to see if there can be anything that can be cancelled out?
    (3 votes)
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    • leaf green style avatar for user kubleeka
      No, because doing that would change the final answer. If, for example, we found a (z+8) factor to cancel, then our final function would be defined at z=-8, while the original is not, so they're different functions.

      But as it happens, the polynomial in the numerator has no linear factors anyway, which you can check with the rational root theorem.
      (3 votes)
  • duskpin tree style avatar for user harrypotter101_ella
    is that number with a line through it a 2 or a 7? Because it looks like a 2 and a 7 mixed together
    (4 votes)
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  • male robot hal style avatar for user basecodeAGTC
    At he inverts the numerator of the second fraction
    3(z+8) becomes -3(z+8). Why is this necessary?

    Edit: There is subtraction involved so I imagine this is to accommodate the fact that the signs change when combining them. If this was addition the sign change would not be necessary. Am I wrong?
    (3 votes)
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    • leaf orange style avatar for user A/V
      That is correct. He did invert the coefficient because he wanted to use the addition operator, so he accommodated the fact that one has to change the signs after distributing the subtraction sign.
      (3 votes)
  • leaf green style avatar for user Mavarez454
    How does Sal know not to factor the numerator(I am referring to the final answer). So he would check for common factors to cancel out. Is there a giveaway in expressions that tell you that the numerator and denominator don't have common factors. Or did Sal do this problem beforehand and factored out the numerator and saw that there were no common factors.
    (2 votes)
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  • starky sapling style avatar for user 🐯🐾tiggress🐯🐾
    How do u find least common denominator with letters and numbers?🤔
    (1 vote)
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  • blobby green style avatar for user holmesshine11
    I still dont get it!! How can you find the LCM if you dont know the denominator in the first place!
    (2 votes)
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  • blobby green style avatar for user Taylor Boney
    Why do you not multiply the 3 with (z+8)(9z-5)(z+6) and only with the (z+8) ?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      Each fraction needs to be converted to the common denominator: (z+8)(9z-5)(z+6). The factor with 3 in the numerator already has the factors of (9z-5)(z+6) in the denominator. So, we multiply the fraction by the factor that is missing: (z+8)/(z+8).

      If you multiplied by (z+8)(9z-5)(z+6), your denominator would become: (9z-5)(z+6)(z+8)(9z-5)(z+6) which is much bigger than the common denominator that you are trying to create.

      Hope this helps.
      (3 votes)

Video transcript

- [Voiceover] Pause this video and see if you can subtract this magenta rational expression from this yellow one. Alright, now let's do this together. And the first thing that jumps out at you is that you realize these don't have the same denominator and you would like them to have the same denominator. And so you might say, well, let me rewrite them so that they have a common denominator. And a common denominator that will work will be one that is divisible by each of these denominators. So it has all the factors of each of these denominators and lucky for us, each of these denominators are already factored. So let me just write the common denominator, I'll start rewriting the yellow expression. So, you have the yellow expression, actually, let me just make it clear, I'm going to write both, the yellow one, and then you're going to subtract the magenta one. Whoops. I'm saying yellow but drawing in magenta. So you have the yellow expression which I'm about to rewrite, actually, I'm going to make a longer line, so the yellow expression minus the magenta one, minus the magenta one, right over there. Now, as I mentioned, we want to have a denominator that has all, the common denominator has to have, be divisible by both, this yellow denominator and this magenta one. So it's got to have the Z plus eight in it. It's got to have the 9z minus five in it. And it's also got to have both of these. Well, I already, we already accounted for the 9z minus five. So it has to have, be divisible by Z plus six. Z plus six. Notice just by multiplying the denominator by Z plus six, we're not divisible by both of these factors AND both of these factors because 9z minus five was the factor common to both of them. And if you were just dealing with numbers when you were just adding or subtracting fractions, it works the exact same way. Alright, so what will the numerator become? Well, we multiply the denominator times Z plus six, so we have to do the same thing to the numerator. It's going to be negative Z to the third times Z plus six. Now let's focus over here. We had, well, we want the same denominator, so we can write this as Z plus eight Z plus eight times Z plus six, times Z plus six times 9z minus five. And these are equivalent. I've just changed the order that we multiply in it, that doesn't change their value. And if we multiplied the, so we had a three on top before and if we multiply the denominator times Z plus eight, we also have to multiply the numerator times Z plus eight. So there you go. And so, this is going to be equal to, this is going to be equal to, actually, I'll just make a big line right over here. This is all going to be equal to. We have our, probably don't need that much space, let me see, maybe that, maybe about that much. So I'm going to have the same denominator and I'll just write it in a neutral color now. Z plus eight times 9z minus five times Z plus six. So over here, just in this blue color, we want to distribute this negative Z to the third. Negative Z to the third times Z is negative Z to the fourth. Negative Z to the third times six is minus 6z to the third. And now this negative sign, right over here, actually, instead of saying negative Z, negative of this entire thing, we could just say plus the negative of this. Or another of thinking about it, you could view this as negative three times Z plus eight. So we could just distribute that. So let's do that. So negative three times Z is negative 3z and negative three times eight is negative 24. And there you go. We are, we are done. We found a common denominator. And once you have a common denominator, you could just subtract or add the numerators, and instead of doing this as minus this entire thing, I viewed it as adding and then having a negative three in the numerator, distributing that and then these, I can't simplify it any further. Sometimes you'll do one of these types of exercises and you might have two second-degree terms or two first-degree terms or two constants or something like that and then you might want to add or subtract them to simplify it but here, these all have different degrees so I can't simplify it any further and so we are all done.