Where defined, the expression (x+2)/((x²+25)/(x²-4)) is equivalent to which of the following?
((x+2)^2(x-2))/((x^2+25)(x-5))
To simplify the expression ((x+2)/((x²+25)/(x²-4))), we first rewrite the denominator as ((x²-4) * (x+2)/(x²+25)). This leads us to the correct answer, which simplifies to (((x+2)^2(x-2))/((x^2+25)(x-5))).
This choice represents a product of two linear factors but does not account for the original expression's dependency on ((x+2)) and the quadratic term in the denominator. Therefore, it fails to maintain the necessary structure of the original expression.
While this choice is a valid rational expression, it does not reflect the original expression's complexity. The lack of ((x+2)) in either the numerator or denominator means it cannot be equivalent to the original expression.
This option presents another rational expression, yet it disregards the required factor of ((x+2)) entirely. The absence of both the quadratic component in the denominator and the necessary multiplicative structure renders this choice incorrect.
This choice has a different structure and does not incorporate the necessary factors of ((x+2)) which are crucial to matching the original expression. The numerator and denominator also differ significantly from the original structure, making it non-equivalent.
The expression ((x+2)/((x²+25)/(x²-4))) simplifies to (((x+2)^2(x-2))/((x^2+25)(x-5))), demonstrating that the correct answer is E. Each incorrect choice fails to maintain the necessary elements from the original expression, highlighting the importance of accurately reflecting the structure and components when simplifying rational expressions.
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