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

Question

Isaiah Anthony · Accepted Answer

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

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

# = (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#

Applying property #a^2- b^2 = (a+b)(a-b)#

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

#= 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) #

Isaiah Anthony · Answer

undefined To simplify the expression (4x^3)/(2x^2) ÷ (x^2-9)/(x^2-4x-21), we can follow these steps:

1. Simplify the numerator of the first fraction: (4x^3)/(2x^2) simplifies to 2x.

2. Simplify the denominator of the first fraction: (x^2-9)/(x^2-4x-21) can be factored as (x-3)(x+3)/[(x-7)(x+3)].

3. Cancel out common factors between the numerator and denominator: The (x+3) terms in both the numerator and denominator can be canceled out.

4. Simplify the remaining expression: After canceling out the common factors, we are left with 2x/[(x-7)].

Therefore, the simplified expression is 2x/(x-7).