Perform the operation and simplify.
$\frac{z^2-4}{z^2-1} \cdot \frac{z+1}{z+2}$
Answer (1)
Factor the numerator: z 2 − 4 = ( z − 2 ) ( z + 2 ) .
Factor the denominator: z 2 − 1 = ( z − 1 ) ( z + 1 ) .
Rewrite the expression: ( z − 1 ) ( z + 1 ) ( z − 2 ) ( z + 2 ) ⋅ z + 2 z + 1 .
Cancel common factors and simplify: z − 1 z − 2 .
The simplified expression is z − 1 z − 2 .
Explanation
Understanding the Problem We are asked to simplify the expression z 2 − 1 z 2 − 4 ⋅ z + 2 z + 1 . This involves factoring, canceling common factors, and simplifying the result.
Factoring the Numerator First, we factor the numerator z 2 − 4 as a difference of squares: z 2 − 4 = ( z − 2 ) ( z + 2 ) .
Factoring the Denominator Next, we factor the denominator z 2 − 1 as a difference of squares: z 2 − 1 = ( z − 1 ) ( z + 1 ) .
Rewriting the Expression Now we rewrite the original expression with the factored forms: z 2 − 1 z 2 − 4 ⋅ z + 2 z + 1 = ( z − 1 ) ( z + 1 ) ( z − 2 ) ( z + 2 ) ⋅ z + 2 z + 1 .
Canceling Common Factors We can now cancel the common factors ( z + 2 ) and ( z + 1 ) from the numerator and the denominator: ( z − 1 ) ( z + 1 ) ( z − 2 ) ( z + 2 ) ⋅ z + 2 z + 1 = ( z − 1 ) ( z + 1 ) ( z − 2 ) ( z + 2 ) ⋅ z + 2 z + 1 = z − 1 z − 2 .
Final Answer Therefore, the simplified expression is z − 1 z − 2 .
Examples
Simplifying rational expressions is a fundamental skill in algebra, useful in various fields such as physics and engineering. For instance, when analyzing electrical circuits, you might encounter complex expressions involving impedances. Simplifying these expressions using techniques like factoring and canceling common factors can make the analysis more manageable and lead to a clearer understanding of the circuit's behavior. Similarly, in physics, simplifying expressions can help in solving problems related to wave mechanics or fluid dynamics.
