Given that `anepm(1)/(2)`, which of the following is equivalent to `(2a^(2)+5a-3)/(4a^(2)-1)`?

Options

A. `(2a-3)/(2a+1)`
B. `(a-3)/(2a-1)`
C. `(2a+3)/(2a-1)`
D. `(a+3)/(2a+1)`

Answer

D

Explanation

Difficult: Medium
Category: Passport to Advanced Math/ Exponents
Strategic advice: When dividing polynomials, you have two choices: factor and cancel, or long division. Choose whichever method you're more comfortable with.
Getting to the Answer: Factoring is most likely the faster route. The denominator is easier to factor, so start there: `4a^(2)-1` is a diffence fo squares which becomes `(2a+1)(2a-1)`. The expression in the numerator will factor into two binomials, one of which must be either 2a+1 or 2a-1 so that the fraction can be reduced. Using factoring y grouping, find the factors of `2(-3)=-6` that add up to +5, split the middle term using those numbers, and then factor by grouping:
`2a^(2)+5a-3=2a^(2)+6a-a-3=2a(a+3)-1(a+3)=(2a-1)(a+3)`
The factor 2a-1 appears in the numerator and the denominator, so cancel it to leave `(a+3)/(2a+1)`
Which is (D).