How do you simplify (x-2)/(x^3-8)x2x38?

2 Answers
May 12, 2018

1/(x^2+2x+4)1x2+2x+4

Explanation:

x^3-8 = (x-2)(x^2+2x+4)x38=(x2)(x2+2x+4)
The above uses the rule x^3-y^3 = (x-y)(x^2+xy+y^2)x3y3=(xy)(x2+xy+y2)

(x-2)/(x^3-8) = (x-2)/((x-2)(x^2+2x+4)) = 1/(x^2+2x+4)x2x38=x2(x2)(x2+2x+4)=1x2+2x+4

May 12, 2018

1/(x^2+2x+4),x!=21x2+2x+4,x2

Explanation:

" "(x-2)/(x^3-8) x2x38

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 x380
" "x^3!=8 x38
" "x!=2 x2

Continue solving by putting 88 into exponent form.
=(x-2)/(x^3-2^3)=x2x323

Factor the difference of cubes (a^3-b^3=(a-b)(a^2+ab+b^2)a3b3=(ab)(a2+ab+b2)).
=(x-2)/((x-2)(x^2+2x+4))=x2(x2)(x2+2x+4)

Reduce the fraction with x-2x2.
=cancel(x-2)/(cancel(x-2)(x^2+2x+4))
=1/(x^2+2x+4),x!=2