Question

1. what is the quotient represented by the expression \\(\frac{8a^2 - 26a + 21}{4a - 7}\\) 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 the numerator

Factor quadratic $8a^2-26a+21$.
We find two numbers: $-14$ and $-12$, since $(-14)\times(-12)=168$ and $-14-12=-26$.
Rewrite and factor:

$$\begin{align*} 8a^2-14a-12a+21&=2a(4a-7)-3(4a-7)\\ &=(2a-3)(4a-7) \end{align*}$$

Step2: Simplify the rational expression

Cancel common factor $4a-7$ (where $4a-7
eq0$).

$$ \frac{(2a-3)(4a-7)}{4a-7}=2a-3 $$

Step3: Define expression's domain

The original expression is undefined when denominator $4a-7=0$, so $a
eq\frac{7}{4}$. The simplified quotient is valid for all $a
eq\frac{7}{4}$.

Answer:

b. $2a - 3$