Which one of the following is a simplification of
[tex]$\frac{(x+2)^2}{x^2+x-2}$[/tex]
A. [tex]$\frac{x+2}{x-1}$[/tex]
B. [tex]$\frac{(x+2)^2}{(x-2)(x+1)}$[/tex]
C. [tex]$\frac{4 x+4}{x-2}$[/tex]
D. [tex]$\frac{x+2}{x-2}$[/tex]
E. -4
Answer (2)
Factor the denominator: x 2 + x − 2 = ( x + 2 ) ( x − 1 ) .
Rewrite the expression: ( x + 2 ) ( x − 1 ) ( x + 2 ) 2 .
Cancel the common factor: ( x + 2 ) ( x − 1 ) ( x + 2 ) 2 = x − 1 x + 2 .
The simplified expression is x − 1 x + 2 .
Explanation
Understanding the problem We are given the expression x 2 + x − 2 ( x + 2 ) 2 and asked to simplify it. To do this, we need to factor the denominator and see if there are any common factors with the numerator that can be cancelled.
Factoring the denominator First, let's factor the denominator x 2 + x − 2 . We are looking for two numbers that multiply to -2 and add to 1. These numbers are 2 and -1. So, we can write the denominator as ( x + 2 ) ( x − 1 ) .
Rewriting the expression Now we can rewrite the original expression as ( x + 2 ) ( x − 1 ) ( x + 2 ) 2 .
Cancelling common factors We can cancel the common factor of ( x + 2 ) from the numerator and the denominator, which gives us x − 1 x + 2 .
Final Answer Therefore, the simplified expression is x − 1 x + 2 . Comparing this to the given options, we see that it matches option A.
Examples
Simplifying rational expressions is a fundamental skill in algebra and is used in many real-world applications. For example, in physics, you might encounter rational expressions when dealing with rates of change or when analyzing the behavior of electrical circuits. In economics, rational expressions can be used to model cost-benefit ratios or to analyze supply and demand curves. By simplifying these expressions, you can gain a better understanding of the underlying relationships and make more informed decisions.
The simplified form of x 2 + x − 2 ( x + 2 ) 2 is x − 1 x + 2 , which corresponds to option A.
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