Question

How do you combine `\frac { 5} { y ^ { 2} + 10y + 21} + \frac { 3y } { y ^ { 2} + 6y - 7} - \frac { 2} { y ^ { 2} + 2y - 3}`?

Answer 1

`(3y^2+12y-19)/((y+7)(y+3)(y-1))`

Explanation:

Let's find a common denominator. We can do that by factoring the denominators and seeing what terms need to be included:

`5/(y^2+10y+21)+(3y)/(y^2+6y-7)-2/(y^2+2y-3)`

`5/((y+7)(y+3))+(3y)/((y+7)(y-1))-2/((y+3)(y-1))`

And so the denominator needs `(y+7)(y+3)(y-1)`:

`5/((y+7)(y+3))(1)+(3y)/((y+7)(y-1))(1)-2/((y+3)(y-1))(1)`

`5/((y+7)(y+3))(y-1)/(y-1)+(3y)/((y+7)(y-1))(y+3)/(y+3)-2/((y+3)(y-1))(y+7)/(y+7)`

`(5(y-1)+3y(y+3)-2(y+7))/((y+7)(y+3)(y-1))`

`(5y-5+3y^2+9y-2y-14)/((y+7)(y+3)(y-1))`

`(3y^2+12y-19)/((y+7)(y+3)(y-1))`

I think the numerator won't factor neatly and so I'll leave it like this.