Question

What is the sum?

StartFraction 3 y Over y squared + 7 y + 10 EndFraction + StartFraction 2 Over y + 2 EndFraction
StartFraction 5 Over y minus 5 EndFraction
StartFraction 5 (y + 2) Over (y minus 2) (y + 5) EndFraction
StartFraction 5 Over y + 5 EndFraction
StartFraction 5 (y minus 2) Over (y minus 5) (y + 2) EndFraction

Answer 1

First factor y^2 + 7y + 10 = (y+5)(y+2). Group convenient terms:

3y/((y+5)(y+2)) + 2/(y+2) = (3y + 2(y+5))/((y+5)(y+2)) = 5(y+2)/((y+5)(y+2)) = 5/(y+5).

So the sum becomes
5/(y+5) + 5/(y-5) + 5(y+2)/((y-2)(y+5)) + 5/(y+5) + 5(y-2)/((y-5)(y+2)).

Combine the two 5/(y+5) terms: 10/(y+5). Now pair and combine:

10/(y+5) + 5(y+2)/((y-2)(y+5)) = 5(3y-2)/((y-2)(y+5)),
and
5/(y-5) + 5(y-2)/((y-5)(y+2)) = 10y/((y-5)(y+2)).

Adding these over the common denominator (y-2)(y+5)(y-5)(y+2) gives the final result

(25y^3 - 25y^2 - 220y + 100)/((y-2)(y+2)(y-5)(y+5)).

You may factor a 5 from the numerator:

5(5y^3 - 5y^2 - 44y + 20)/((y-2)(y+2)(y-5)(y+5)),

with y ≠ ±2, ±5.