Question

What is the quotient?

StartFraction 2 y squared minus 6 y minus 20 Over 4 y + 12 EndFraction divided by StartFraction y squared + 5 y + 6 Over 3 y squared + 18 y + 27 EndFraction
StartFraction 2 Over 3 (y minus 5) EndFraction
StartFraction 3 (y minus 5) Over 2 EndFraction
StartFraction (y minus 5) (y + 2) squared Over 6 (y + 3) squared EndFraction
StartFraction 3 (y + 5) (y minus 2) Over 2 (y + 2) EndFraction

Answer 1

Compute and simplify:

1. Factor each polynomial:
- 2y^2 - 6y - 20 = 2(y - 5)(y + 2)
- 4y + 12 = 4(y + 3)
- y^2 + 5y + 6 = (y + 2)(y + 3)
- 3y^2 + 18y + 27 = 3(y + 3)^2

2. Form the division and use reciprocal:
[(2(y-5)(y+2))/(4(y+3))] ÷ [(y+2)(y+3)/(3(y+3)^2)]
= [(y-5)(y+2)/(2(y+3))] * [3(y+3)/(y+2)]

3. Cancel (y+2) and (y+3):
= 3(y-5)/2.

So the quotient is 3(y - 5)/2 (second choice). Exclude y = -3 and y = -2 (division undefined).