Solve g(y)=y^3+6y/y^2-36divy^2+y-56/y^2+2y-48

$$g ( y ) = \frac { y ^ { 3 } + 6 y } { y ^ { 2 } - 36 } \div \frac { y ^ { 2 } + y - 56 } { y ^ { 2 } + 2 y - 48 }$$

Evaluate

$$\frac{y\left(y^{2}+6\right)}{\left(y-7\right)\left(y+6\right)}$$

Short Solution Steps

Divide \frac{y^{3}+6y}{y^{2}-36} by \frac{y^{2}+y-56}{y^{2}+2y-48} by multiplying \frac{y^{3}+6y}{y^{2}-36} by the reciprocal of \frac{y^{2}+y-56}{y^{2}+2y-48}.

$$\frac{\left(y^{3}+6y\right)\left(y^{2}+2y-48\right)}{\left(y^{2}-36\right)\left(y^{2}+y-56\right)}$$

Factor the expressions that are not already factored.

$$\frac{y\left(y-6\right)\left(y+8\right)\left(y^{2}+6\right)}{\left(y-7\right)\left(y-6\right)\left(y+6\right)\left(y+8\right)}$$

Cancel out \left(y-6\right)\left(y+8\right) in both numerator and denominator.

$$\frac{y\left(y^{2}+6\right)}{\left(y-7\right)\left(y+6\right)}$$

Expand the expression.

$$\frac{y^{3}+6y}{y^{2}-y-42}$$

Expand

$$\frac{y^{3}+6y}{\left(y-7\right)\left(y+6\right)}$$

Short Solution Steps

Divide \frac{y^{3}+6y}{y^{2}-36} by \frac{y^{2}+y-56}{y^{2}+2y-48} by multiplying \frac{y^{3}+6y}{y^{2}-36} by the reciprocal of \frac{y^{2}+y-56}{y^{2}+2y-48}.

$$\frac{\left(y^{3}+6y\right)\left(y^{2}+2y-48\right)}{\left(y^{2}-36\right)\left(y^{2}+y-56\right)}$$

Factor the expressions that are not already factored.

$$\frac{y\left(y-6\right)\left(y+8\right)\left(y^{2}+6\right)}{\left(y-7\right)\left(y-6\right)\left(y+6\right)\left(y+8\right)}$$

Cancel out \left(y-6\right)\left(y+8\right) in both numerator and denominator.

$$\frac{y\left(y^{2}+6\right)}{\left(y-7\right)\left(y+6\right)}$$

Expand the expression.

$$\frac{y^{3}+6y}{y^{2}-y-42}$$