Main content

## Precalculus

### Course: Precalculus > Unit 4

Lesson 7: Adding and subtracting rational expressions- Intro to adding & subtracting rational expressions
- Adding & subtracting rational expressions: like denominators
- Intro to adding rational expressions with unlike denominators
- Adding rational expression: unlike denominators
- Subtracting rational expressions: unlike denominators
- Add & subtract rational expressions (basic)
- Adding & subtracting rational expressions
- Least common multiple of polynomials
- Subtracting rational expressions: factored denominators
- Subtracting rational expressions
- Add & subtract rational expressions
- Rational functions: FAQ

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Adding & subtracting rational expressions: like denominators

Sal adds 6/(2x²-7) + (-3x-8)/(2x²-7) and subtracts (9x²+3)/(14x²-9) - (-3x²+5)/(14x²-9).

## Want to join the conversation?

- Can you simplify this any further:

2Y+7 all over 2Y

I would assume that you can't since the +7 is in the numerator but I'm not extremely confident(12 votes)- You are correct... you can't simplify any further. The +7 is preventing any further reduction. We can only cancel factors (items being multiplied). Since the numerator contains addition, it prevents us from cancelling.(15 votes)

- In the second problem,why do we distribute a negative sign and why do we distribute? If we do it the first method as well,will we get the same answer as distributing?(7 votes)
- you distribute the negative sign because you are subtracting. If you don't distribute then you would be adding.(1 vote)

- what is the product of 5^4 and 5^4(3 votes)
- These two objects have the same base. So when you multiply them, you can add the exponents. 5^4×5^4=5^(4+4)=5^8.(8 votes)

- Given a rational expression, identify the excluded values by finding the zeroes of the denominator.(6 votes)
- square root of 20/x^2 + square root of 5/4x^2 =(6 votes)
- Its simple you just have to add the 2 2y's and leave the seven, See Simple.(4 votes)
- At3:36, why can't you simplify 12x^2 and 14x^2 by 2 to be 6x^2 - 2 / 4x^2 - 9?(2 votes)
- No, we can't simplify any further. When we reduce fractions we cancel out common factors (items being multiplied. For example: 10/15 = 2/3 because 10 = 2*5 and 15 = 3*5 and they share a common factor of 5. In the video, 12x^2 and 14x^2 are terms (they are being added/subtracted with other values). We can't cancel terms.

Hope this helps.(5 votes)

- On video at1:18why is it -2-3x and not -3x-2 instead? Many thanks in advance.(2 votes)
- The 2 expressions are equivalent. "-3x-2" would be preferred as it is in standard form. But, it isn't required.(4 votes)

- How about if we have like numerators, do we take the common factor too?(2 votes)
- When adding/subtracting all fractions, we need a common denominator. There is no requirement to have a common numerator.(4 votes)

- (-4-4x)/x^2-x-2. How do you simplify this?(2 votes)
- Factor both the top and the bottom. Factor out the GCF (-4) from the top and factor the bottom into two binomials:

-4(x + 1)/[(x+1)(x-2)]

Then cancel the common factor to get -4/(x-2).(3 votes)

## Video transcript

- [Voiceover] So let's add six over two X squared minus seven to negative 3 X minus eight over two X squared minus seven. And like always, pause the video and try to work it out before I do. When you look at this, we have these two rational expressions and we have the same denominator, two X squared minus seven. So you could say, we have six
two X squared minus sevenths and then we have negative
three X minus eight two X squared minus sevenths
is one way to think about it. So if you have the same denominator, this is going to be equal to, this is going to be equal to... our denominator is going to
be two X squared minus seven, two X squared minus seven, and then we just add the numerators. So it's going to be six
plus negative three X, negative three X minus eight. So if we want to simplify
this a little bit, we'd recognize that we can
add these two constant terms, the six and the negative eight. Six plus negative eight is going to be negative two, so it's
going to be negative two and then adding a negative three X, that's the same thing
as subtracting three X, so negative two minus
three X, all of that over, all of that with that same blue color, all of that over two X squared minus seven. And we're done. We've just added these
two rational expressions. Let's do another example. So here, we want to subtract one rational expression from another. So see if you can figure that out. Well, once again, both of
these rational expressions have the exact same denominator, the denominator for both of them is 14 X squared minus nine,
14 X squared minus nine. So the denominator of the difference, I guess we can call it that, is going to be 14 X squared minus nine. So 14 X squared minus nine. Did I say four X squared before? 14 X squared minus nine, that's the denominator of both of them, so that's going to be the denominator of our answer right over here. And so, we can just
subtract the numerators. So we're gonna have nine X squared plus three minus all of this business, minus negative three X squared plus five. And so we can distribute
the negative sign. This is going to be equal to nine X squared plus three, and then, if you distribute the negative sign, the negative of negative three X squared is going to be plus three X squared and then the negative of positive five is going to be negative five, so we're gonna subtract five from that, and all of that is going to be
over 14 X squared minus nine. 14 X squared minus nine. And so in the numerator we
can do some simplification. We have nine X squared
plus three X squared, so that's going to be
equal to 12 X squared. And then, we have... we have three plus negative five, or we can say three minus five, so that's going to be negative two, and all of that is going to be over 14 X squared minus nine. 14 X squared minus nine. And we're all done. We have just subtracted. And we can think about it, is there any way we
can simplify this more, are there any common factors, but these both could be
considered differences of squares, but they're going to be
differences of squares of different things, so they're not going
to have common factors. So this is about as simple as we can get.