Solve M(z)=z-1/z+1z^2/(z-1)(z+1)=

$$M ( z ) = \frac { z - 1 } { z + 1 } \frac { z ^ { 2 } } { ( z - 1 ) ( z + 1 ) } =$$

Evaluate

$$\left(\frac{z}{z+1}\right)^{2}$$

Short Solution Steps

Consider \left(z-1\right)\left(z+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$\frac{z-1}{z+1}\times \frac{z^{2}}{z^{2}-1}$$

Multiply \frac{z-1}{z+1} times \frac{z^{2}}{z^{2}-1} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(z-1\right)z^{2}}{\left(z+1\right)\left(z^{2}-1\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(z-1\right)z^{2}}{\left(z-1\right)\left(z+1\right)^{2}}$$

Cancel out z-1 in both numerator and denominator.

$$\frac{z^{2}}{\left(z+1\right)^{2}}$$

Expand the expression.

$$\frac{z^{2}}{z^{2}+2z+1}$$

Expand

$$\left(\frac{z}{z+1}\right)^{2}$$

Short Solution Steps

Consider \left(z-1\right)\left(z+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$\frac{z-1}{z+1}\times \frac{z^{2}}{z^{2}-1}$$

Multiply \frac{z-1}{z+1} times \frac{z^{2}}{z^{2}-1} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(z-1\right)z^{2}}{\left(z+1\right)\left(z^{2}-1\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(z-1\right)z^{2}}{\left(z-1\right)\left(z+1\right)^{2}}$$

Cancel out z-1 in both numerator and denominator.

$$\frac{z^{2}}{\left(z+1\right)^{2}}$$

Expand the expression.

$$\frac{z^{2}}{z^{2}+2z+1}$$