Question

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

Answer 1

`= (2 x^2- 14x )/ (x-3)`

Explanation:

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

• Simplifying `color(blue)((4x^3) / (2x^2) ):`

`= (4/ 2 ) * 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 = (a+b)(a-b)`

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

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

`= x ( x- 7 ) + 3 (x - 7)`

`= 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 )`

`= color(blue)( 2x) xx color(purple)((x+3) ( x- 7 ) )/ color(green)((x+3)(x-3))`

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

`= (2x xx ( x- 7 ) )/ (x-3)`

`= (2 x^2- 14x )/ (x-3)`