31. Where defined, which of the following is equivalent to (4-25x²)/(5x²-13x-6)?
Answer: A
(2-5x)/(x-3)
The expression (4-25x²)/(5x²-13x-6) simplifies to (2-5x)/(x-3) when the numerator and denominator are factored appropriately.
A) (2-5x)/(x-3)
This option is correct as it represents the simplified form of the given expression. By factoring both the numerator and the denominator, it can be shown that (4-25x²) = -(5)(x-3)(2-5x) and (5x²-13x-6) = (x-3)(5x+2), leading to the cancellation of (x-3) and yielding (2-5x)/(x-3).
B) (5x-2)/(x-3)
This option is incorrect because it does not represent the simplified form of the original expression. The factors do not align with (4-25x²) or (5x²-13x-6) once they are factored, as shown in the correct answer.
C) (2+5x)/(x-3)
This option is also incorrect because it alters the sign of the term involving x in the numerator. The correct simplification must maintain the original structure of (4-25x²), which does not lead to a positive 5x.
D) (2-5x)/(x+3)
This option is incorrect since it changes the sign in the denominator. The original expression’s denominator (5x²-13x-6) does not factor to (x+3), thus making this equivalence invalid.
E) (5x-2)/(x+3)
This option is incorrect as well. Similar to option D, it misrepresents the factors of the denominator and does not simplify correctly from the original expression.
Conclusion
The simplification of (4-25x²)/(5x²-13x-6) accurately leads to (2-5x)/(x-3), making option A the only correct choice. All other options either misrepresent the factors or alter the signs, thus failing to maintain equivalence with the original expression.