# How do you simplify the rational expression: (x+2)/(4x-8)(3x-9)/(x+4)(2x-21)/(x^2-x-6)?

Oct 14, 2015

$\frac{6 x - 63}{4 {x}^{2} + 8 x - 32}$

#### Explanation:

The idea is to factor out numbers is the first-degree polynomial (if possible), and to factor the highest-degree, finding their roots (if possible). So, let's work separately on the three pieces:

• First fraction:
Numerator: $x + 2 \to$ nothing to do;
**Denominator: $4 x - 8 \to$ can factor a $4$, obtaining $4 \left(x - 2\right)$.

• Second fraction:
Numerator: $3 x - 9 \to$ can factor a $3$, obtaining $3 \left(x - 3\right)$;
**Denominator: $x + 4 \to$ nothing to do.

• First fraction:
Numerator: $2 x - 21 \to$ nothing to do;
**Denominator: ${x}^{2} - x - 6 \to$ its roots are $3$ and $- 2$, so we can write it as $\left(x - 3\right) \left(x + 2\right)$.

Writing back the whole expression with this changes gives

$\frac{\textcolor{red}{\cancel{x + 2}}}{4 \left(x - 2\right)} \cdot \frac{3 \textcolor{b l u e}{\cancel{\left(x - 3\right)}}}{x + 4} \cdot \frac{2 x - 21}{\textcolor{b l u e}{\cancel{\left(x - 3\right)}} \textcolor{red}{\cancel{\left(x + 2\right)}}}$