Rewrite the rational expression as an equivalent rational expression Y over y3+6y2+5y = blank over y (y+5)(y+1)(y+2)

Solution:

1. Start with the given rational expression:

\frac{Y}{y^3+6y^2+5y}

2. Factor the denominator:

* Factor out the common factor y :

y^3 + 6y^2 + 5y = y(y^2 + 6y + 5)

* Factor the quadratic y^2 + 6y + 5 :

y^2 + 6y + 5 = (y + 5)(y + 1)

3. Substitute the factored form into the expression:

\frac{Y}{y(y+5)(y+1)}

4. Compare with the desired form:

y(y + 5)(y + 1)(y + 2)

5. Solve for Y in the equivalent expression with the given denominator:

* The missing factor is y + 2 , multiply the numerator by this factor:

* Find Y , multiply the current numerator by the missing factor:

Y = Y(y + 2)

6. Hence, the equivalent rational expression is:

\frac{Y(y + 2)}{y(y + 5)(y + 1)(y + 2)}