In Exercises 33–68, add or subtract as indicated. (4x^2+x−6)/(x^2...

Question

In Exercises 33–68, add or subtract as indicated. (4x^2+x−6)/(x^2+3x+2)−3x/(x+1)+5/(x+2)

Accepted Answer

hey everyone here we are asked to add or subtract as indicated in the expression three X squared plus X divided by X squared plus nine X plus 14 minus three X divided by X plus two plus one divided by X plus seven. Now to be able to add and subtract fractions, we need to have a common denominator but before we can start changing these denominators to make them the same, we want to make sure that they are each fully factored out. So starting with the first expression we have three X squared plus X in the numerator. But we see that the denominator has a trine Amiel of which can be factored out into the quantity of X plus two times the quantity of X plus seven. Now we can move on and take a look at the 2nd and 3rd expressions and we see that their denominators are fully factored but we want to note that one fraction has X plus two in the denominator and the other has X plus seven. And both of these are already included in our first expression here. So we know that our least common denominator is the quantity of X plus two times the quantity of X plus seven. So now we can move on to getting these second two expressions to have that common denominator. So taking a look at our second expression here of three X divided by X plus two. We can see that it's missing that X plus seven. So we can just multiply both the numerator and the denominator by X plus seven. This then gives us three X times the quantity of X plus seven divided by the quantity of X plus two times the quantity of X plus seven. And now just factoring in that three X. In the numerator we have three X squared plus 21 X divided by the quantity of X plus two divided time. Sorry the quantity of X plus seven. Now we're all done here we can move on to the third expression of one divided by X plus seven. We see that it's missing that X plus two. So just repeating the same idea. We're gonna multiply both the numerator and the denominator by X plus two to get the common denominator and this gives us X plus two divided by the quantity of X plus two times the quantity of X plus seven. Now we have each of the expressions with the same denominator. So we can start combining the enumerators. So our first expression gives us three X squared plus X. Our second gives us minus three X squared plus 21 X. And finally the third plus X plus two. All divided by the quantity of X plus two times the quantity of X plus seven. Now I'm just gonna factor in this negative sign here in the numerator. So we can see exactly what we're dealing with. So we have three X squared plus X minus three X squared minus 21 X plus X plus two divided by the quantity of X plus two times the quantity of X plus seven. And from here we can just start simplifying. We see that the three X squares cancel out. So just adding everything up in the numerator we have negative 19 X plus two, divided by the quantity of X plus two times the quantity of X plus seven. And this is our answer here for choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 33–68, add or subtract as indicated. (4x^2+x−6)/(x^2+3x+2)−3x/(x+1)+5/(x+2)

Key Concepts

Rational Expressions

Finding a Common Denominator

Polynomial Factoring

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