in this lesson we're going to add and subtract rational expressions with unlike denominators so let's start with this example 5 over x plus 3 divided by x squared now before we can combine the two fractions we need to get common denominators so we need to multiply the first fraction on the left by x over x so that the denominators will both be x squared so this is 5x over x squared plus 3 divided by x squared so now we can write this as a single fraction 5x plus 3 over x squared so that's going to be the answer let's try another example x minus 3 divided by 4 minus x plus 2 divided by 3. so we need to get common denominators the least common multiple of four and three is twelve if you're not sure just multiply four and three any multiple of four three will work you may have to simplify it later but it can help you get the right answer let's multiply the first fraction by three and the second one by four now let's distribute three to x minus three so it's going to be three x minus nine and three times four is twelve now let's distribute four to x plus two so that's going to be four x plus eight and keep in mind this negative sign needs to be distributed to the 4x and the 8. so now let's write it as a single fraction so we have 3x minus 9 and if we distribute the negative sign it's going to be negative 4x minus 8. so 3x minus 4x is negative x negative 9 minus 8 is negative 17. so if we want to we can take out a negative sign so negative 1 and then it's going to be x plus 17 over 12. so we can write the final answer as negative x plus 17 over 12. so this negative sign you can put it here if you want to so this is the answer now let's move on to our third example 4 divided by x minus 2 plus 5 over x plus 2. so we need to get common denominators we need to multiply this fraction by this denominator and the fraction on the right by the other denominator so let's put x plus 2 on top and on the bottom here and for this one we're going to multiply by x minus 2 top and bottom so let's distribute 4 to x plus 2 so it's going to be four x plus eight and let's go ahead and foil x minus two times x plus two which we can see that it's x squared minus four which is less to write than x minus two then times x plus 2. now let's distribute 5 to x minus 2. it's going to be 5x minus 10. now that we have the same denominator we can add the numerators 4x plus 5x is 9x 8 plus negative 10 is negative 2. so this is the answer and if we want to we can write it like this in its complete factored form and so that's it for this problem here's another one that you could try x divided by x squared plus 9x plus 20. minus 4 over x squared plus 7x plus 12. now before we can get common denominators we should factor completely so what two numbers multiply to 20 but add to nine this is going to be four and five four plus five is nine four times five is twenty now what two numbers multiply to 12 but add to seven this is going to be positive three and positive four so now we need to get common denominators both fractions have an x plus four so we don't have to worry about it the fraction on the right does not have an x plus five so we need to multiply top and bottom by x plus five the fraction on the left does not have x plus three so we need to multiply top and bottom by x plus three now notice that both fractions have the same denominator x plus three x plus four and x plus five so we can now write it as a single fraction so let's distribute x and x plus three that's going to be x squared plus 3x now 4 times x plus 5 is 4x plus 20 but let's incorporate the negative sign so it's going to be negative 4x minus 20. now let's combine like terms 3x minus 4x is negative x and so this is what we currently have now is there anything else that we can do at this point can we factor x squared minus x minus 20 what two numbers multiply to negative 20 but add to negative one this is negative five and plus four negative five plus four is negative one but they multiply it's negative twenty so we can factor as x minus five times x plus four so notice that we can cancel x plus four therefore the final answer is x minus five divided by x plus three times x plus five so that's the solution to this problem let's work on this example x squared over x minus four plus seven over four minus x let's add these two rational expressions so what we need to do is get common denominators but we don't need to multiply or divide by anything we do need to factor out a negative one if we take out negative one negative x will become positive x and positive four will become negative four now this negative we can move to the top so we can rewrite it as negative seven or just minus seven over x minus four so now we can combine the two fractions this is going to be x squared minus 7 over x minus 4 and that's the answer now let's try one last example 5 divided by x plus two plus two divided by x plus one minus three over x minus one so we need to multiply the first fraction by the other two denominators so that's going to be x plus one times x minus one now we need to multiply the second fraction by the other two denominators so that is x plus two times x minus one top and bottom the last fraction we're going to multiply by x plus 2 and x plus 1. now notice that we're going to have the same denominator which is going to be x plus 2 times x plus 1 times x minus 1 so we're going to combine everything into a single fraction so here we have 5 times x plus 1 x minus 1. if we foil x plus 1 times x minus 1 it's going to be x squared minus 1. and then if we foil x plus 2 times x minus 1 it's going to be x squared plus x minus two now let's foil x plus two times x plus one that's going to give you x squared plus three x plus two so now let's distribute five to x squared minus one that's going to be five x squared minus five and now let's distribute the two so it's gonna be plus 2x squared plus 2x minus 4. and let's do the same for the last one it's going to be a negative 3x squared minus 9x minus 6 all divided by the common denominator now let's clear away a few stuff so now what we need to do is combine like terms five plus two is seven and seven minus three is four so we're going to have 4x squared now 2x minus 9x that's going to be negative 7x and then we have negative 5 minus 4 which is negative 9 minus 6 that's going to be negative 15 divided by this stuff so can we factor 4x squared minus 7x minus 15 what would you say well let's find out 4 times negative 15 is negative 60. what two numbers multiply to negative 60 but add to negative seven so 1 and 60 won't work 2 and 30 they're too far apart 3 and 20 not going to work either there's 4 and negative 15. 5 and 12 that could work 5 times negative 12 is negative 60 but 5 plus negative 12 adds up to negative 7. so let's replace negative 7x with negative 12x plus 5x now let's factor by grouping so let's take out the gcf from the first two terms which is 4x 4x squared divided by 4x is x negative 12x divided by 4x is negative 3. and now the gcf in the last two is five inside the first parenthesis is going to be x minus three and then inside the second one we're going to have the stuff on the outside four x plus five divided by x plus two times x plus one times x minus one so there's really nothing we can cancel so this is the answer in its complete factored form there's nothing else that we can do and so that's it for this lesson