How do you simplify #\frac { z } { ( z - 1) ^ { 2} } - \frac { 1} { ( z - 1) ( z + 3) }#?

Question

Savannah Anderson · Accepted Answer

#(z+1)^2/((z-1)^2(z+3))#

#z/(z-1)^2-1/((z-1)(z+3))#

To get a common denminatro, we multiply the numerator and the denominator of the first term by #(z+3)# and those of the second term y #(z-1)#,

#=(z(z+3))/((z-1)^2(z+3))-(z-1)/((z-1)^2(z+3))#

By combining them together,

#=(z^2+3z-(z-1))/((z-1)^2(z+3))#

By simplifying the numerator,

#=(z^2+2z+1)/((z-1)^2(z+3))#

By completing the square of the numerator,

#=(z+1)^2/((z-1)^2(z+3))#

I hope that this was clear.

Jameson Allen · Answer

undefined To simplify the expression $\frac { z } { ( z - 1) ^ { 2} } - \frac { 1} { ( z - 1) ( z + 3) }$, you need to find a common denominator and then combine the fractions.

First, find the least common denominator (LCD) of the two fractions, which is $(z - 1)^2(z + 3)$.

Then, rewrite each fraction with the LCD:

$$
\frac { z } { ( z - 1) ^ { 2} } = \frac { z(z + 3) } { ( z - 1) ^ { 2} (z + 3) }
$$

$$
\frac { 1} { ( z - 1) ( z + 3) } = \frac { (z - 1) ^ { 2} } { ( z - 1) ^ { 2} (z + 3) }
$$

Now, the expression becomes:

$$
\frac { z(z + 3) } { ( z - 1) ^ { 2} (z + 3) } - \frac { (z - 1) ^ { 2} } { ( z - 1) ^ { 2} (z + 3) }
$$

Combine the fractions:

$$
\frac { z(z + 3) - (z - 1) ^ { 2} } { ( z - 1) ^ { 2} (z + 3) }
$$

Expanding the numerator:

$$
\frac { z^2 + 3z - (z^2 - 2z + 1) } { ( z - 1) ^ { 2} (z + 3) }
$$

$$
\frac { z^2 + 3z - z^2 + 2z - 1 } { ( z - 1) ^ { 2} (z + 3) }
$$

$$
\frac { 5z - 1 } { ( z - 1) ^ { 2} (z + 3) }
$$

So, the simplified expression is $\frac { 5z - 1 } { ( z - 1) ^ { 2} (z + 3) }$.