# How do you simplify \frac { 3x ^ { 2} + 2x - 1} { 3x ^ { 2} + x - 2}?

Jul 16, 2018

$\frac{3 x - 1}{3 x - 2}$

#### Explanation:

$\text{to simplify factor the numerator/denominator and cancel}$
$\text{common factors}$

$\textcolor{b l u e}{\text{Numerator}}$

$\text{the factors of the product } 3 \times - 1 = - 3$

$\text{which sum to "+2" are "+3" and } - 1$

$\text{split the middle term using these factors}$

$3 {x}^{2} + 3 x - x - 1 \leftarrow \textcolor{b l u e}{\text{factor by grouping}}$

$= 3 x \left(x + 1\right) - 1 \left(x + 1\right)$

$= \left(x + 1\right) \left(3 x - 1\right)$

$\textcolor{b l u e}{\text{Denominator}}$

$\text{the factors of the product } 3 \times - 2 = - 6$

$\text{which sum to "+1" are "+3" and } - 2$

$3 {x}^{2} + 3 x - 2 x - 2$

$= 3 x \left(x + 1\right) - 2 \left(x + 1\right)$

$= \left(x + 1\right) \left(3 x - 2\right)$

$\frac{3 {x}^{2} + 2 x - 1}{3 {x}^{2} + x - 2}$

$= \frac{\cancel{\left(x + 1\right)} \left(3 x - 1\right)}{\cancel{\left(x + 1\right)} \left(3 x - 2\right)} = \frac{3 x - 1}{3 x - 2}$

$\text{with restriction } x \ne \frac{2}{3}$