Question

1. what is the quotient represented by the expression \\(\frac{16a^2 - 28a + 21}{4a - 3}\\) for all values of \\(a\\) where the expression is defined?

a. \\(2a + 3\\)
b. \\(2a - 3\\)
c. \\(a - 12\\)
d. \\(a + 12\\)

Explanation:

Step1: Factor numerator

First, factor the quadratic numerator $8a^2 - 26a + 21$. We look for two binomials that multiply to this:
$$8a^2 - 26a + 21 = (4a - 7)(2a - 3)$$

Step2: Rewrite and simplify expression

Substitute the factored form into the original expression, then cancel the common non-zero factor $(2a - 3)$:
$$\frac{(4a - 7)(2a - 3)}{2a - 3} = 4a - 7$$
(Note: The expression is undefined when $2a - 3 = 0$, or $a=\frac{3}{2}$, but the question asks for the quotient where the expression is defined.)

Answer:

$\boldsymbol{4a - 7}$