How do you simplify #(x-2)/(x^3-8)#?

2 Answers
May 12, 2018

Answer:

#1/(x^2+2x+4)#

Explanation:

#x^3-8 = (x-2)(x^2+2x+4)#
The above uses the rule #x^3-y^3 = (x-y)(x^2+xy+y^2)#

#(x-2)/(x^3-8) = (x-2)/((x-2)(x^2+2x+4)) = 1/(x^2+2x+4)#

May 12, 2018

Answer:

#1/(x^2+2x+4),x!=2#

Explanation:

#" "(x-2)/(x^3-8)#

In rational expressions, any variable on the denominator must not equal zero, so we can find restrictions from the original equation initially to save time in the end:
#" "x^3-8!=0#
#" "x^3!=8#
#" "x!=2#

Continue solving by putting #8# into exponent form.
#=(x-2)/(x^3-2^3)#

Factor the difference of cubes (#a^3-b^3=(a-b)(a^2+ab+b^2)#).
#=(x-2)/((x-2)(x^2+2x+4))#

Reduce the fraction with #x-2#.
#=cancel(x-2)/(cancel(x-2)(x^2+2x+4))#
#=1/(x^2+2x+4),x!=2#