How do you simplify #(x^2+8x+16)/(x+2)div(x^2+6x+8)/(x^2-4)#?

2 Answers
Mar 4, 2018

Answer:

The factored expression is #(x^2+2x-8)/(x+2)#.

Explanation:

Here's the strategy:

First, factor all the polynomials. Then, cancel any common terms. Next, change the division of two fractions to multiplication, and flip the second fraction. Lastly, multiply the fractions and cancel any factors in common.

#color(white)=(x^2+8x+16)/(x+2)div(x^2+6x+8)/(x^2-4)#

#=((x+4)(x+4))/(x+2)div((x+2)(x+4))/((x-2)(x+2))#

#=((x+4)^2)/(x+2)div(color(red)cancelcolor(black)((x+2))(x+4))/((x-2)color(red)cancelcolor(black)((x+2)))#

#=((x+4)^2)/(x+2)div(x+4)/(x-2)#

#=((x+4)^2)/(x+2)xx(x-2)/(x+4)#

#=((x+4)^color(red)cancelcolor(black)2(x-2))/((x+2)color(red)cancelcolor(black)((x+4)))#

#=((x+4)(x-2))/(x+2)#

#=(x^2+2x-8)/(x+2)#

Here's what the graph looks like:

graph{(x^2+2x-8)/(x+2) [-30.54, 30.53, -28.27, 28.27]}

Mar 4, 2018

Answer:

#((x+4)(x-2))/((x+2))#

Explanation:

#(x^2+8x+16)/(x+2)div(x^2+6x+8)/(x^2-4)#

To divide by a fraction, multiply by the reciprocal:

#color(blue)((x^2+8x+16))/(x+2)xx(color(red)(x^2-4))/color(green)((x^2+6x+8))#

You cannot cancel with fractions unless there are factors.

Factorise wherever possible:

#color(blue)((x+4)(x+4))/(x+2) xx color(red)((x+2)(x-2))/color(green)((x+4)(x+2))#

Cancel common factors

#color(blue)(cancel((x+4))(x+4))/cancel((x+2)) xx color(red)(cancel((x+2))(x-2))/color(green)(cancel((x+4))(x+2))#

#=((x+4)(x-2))/((x+2))#