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

By combining them together,

By simplifying the numerator,

By completing the square of the numerator,

I hope that this was clear.

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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) }).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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