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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))#

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.

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Answer 2

Cora Alexander

To combine the given fractions, first factor the denominators and find their least common denominator. Then, rewrite each fraction with the common denominator and combine the numerators. Finally, simplify the resulting expression if possible.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.