# How do you simplify (4x^3 )/(2x^2)div (x^2-9)/( x^2-4x-21)?

Mar 25, 2016

$= \frac{2 {x}^{2} - 14 x}{x - 3}$

#### Explanation:

color(blue)((4x^3) / (2x^2) ) -: color(green)( (x^2 - 9)) / color(purple)( (x^2 - 4x - 21)

• Simplifying $\textcolor{b l u e}{\frac{4 {x}^{3}}{2 {x}^{2}}} :$

$= \left(\frac{4}{2}\right) \cdot {x}^{3} / {x}^{2}$
As per property : ${a}^{m} / {a}^{n} = {a}^{m - n}$

 = (cancel4/ cancel2 ) * x^ ((3 -2) ) = 2 * x^ 1 = color(blue)(2x

• Factorising color(green)(x^2 - 9 = x^2 - 3^2:

Applying property ${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$

x^2 - 3^2 = color(green)((x+3)(x-3)

• Factorising color(purple)( (x^2 - 4x - 21): (by splitting the middle term)
${x}^{2} - 4 x - 21 = {x}^{2} - 7 x + 3 x - 21$

$= x \left(x - 7\right) + 3 \left(x - 7\right)$

= color(purple)((x+3) ( x- 7 )

The overall expression now becomes:

color(blue)((4x^3) / (2x^2) ) -: color(green)( (x^2 - 9)) / color(purple)( (x^2 - 4x - 21)) = color(blue)( 2x) -: color(green)((x+3)(x-3))/ color(purple)((x+3) ( x- 7 )

$= \textcolor{b l u e}{2 x} \times \frac{\textcolor{p u r p \le}{\left(x + 3\right) \left(x - 7\right)}}{\textcolor{g r e e n}{\left(x + 3\right) \left(x - 3\right)}}$

 = 2x xx (cancel(x+3) ( x- 7 ) )/ (cancel(x+3)(x-3)

$= \frac{2 x \times \left(x - 7\right)}{x - 3}$

$= \frac{2 {x}^{2} - 14 x}{x - 3}$