How do you simplify #\frac { x ^ { 2} } { ( x - 6) ^ { 2} ( x + 1) }#?

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Answer 1 · 2018-08-03T00:31:45

#" "#
#color(red)(\frac { x ^ { 2} } { ( x - 6) ^ { 2} ( x + 1) }=x^2/(x^3-11x^2+24x+36)#

#" "#
We have the rational expression:

#color(blue)(\frac { x ^ { 2} } { ( x - 6) ^ { 2} ( x + 1) }#

Using the identity: #color(green)((a-b)^2 -= (a^2-2ab+b^2)#

#rArr x^2/[(x^2-12x+36)*(x+1)]#

Multiply both the factors at the denominator:

#rArr [(x^2-12x+36)*(x+1)]#

#rArr x(x^2-12x+36)+1(x^2-12x+36)#

#rArr x^3-12x^2+36x+x^2-12x+36#

Combine the like terms and simplify:

#rArr x^3-12x^2+x^2+36x-12x+36#

#rArr x^3-11x^2+24x+36#

In the next step, write the numerator with the above simplified denominator:

#rArr x^2/[x^3-11x^2+24x+36]#

Hence,

#color(green)(\frac { x ^ { 2} } { ( x - 6) ^ { 2} ( x + 1) }=x^2/(x^3-11x^2+24x+36)#

Hope this helps.