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

Question

Eva Adamson · Accepted Answer

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

#(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))#

Avery Alexander · Answer

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

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]}

Ethan Adkins · Answer

undefined To simplify the expression, we can start by factoring the numerator and denominator separately.

The numerator (x^2 + 8x + 16) can be factored as (x + 4)(x + 4), or (x + 4)^2.

The denominator (x + 2) can be factored as (x + 2).

The second fraction's numerator (x^2 + 6x + 8) can be factored as (x + 2)(x + 4).

The second fraction's denominator (x^2 - 4) can be factored as (x + 2)(x - 2).

Now, we can rewrite the expression as (x + 4)^2 / (x + 2) * (x + 2)(x - 2) / (x + 2).

Next, we can cancel out the common factors in the numerator and denominator.

(x + 4)^2 / (x + 2) * (x - 2) / 1.

Finally, we simplify further by multiplying the remaining terms.

(x + 4)^2 * (x - 2) / (x + 2).