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

1 Answer
Mar 25, 2016

Answer:

# = (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) #