How do you simplify #(x^3 − 3x^2 + 3x − 9)/(x^4 - 81)#?

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Answer 1 · 2015-06-14T14:42:51

# = color(red)( (x^2 +3)/ color(blue)((x^2+9)(x+3)#

#color(red)((x^3 − 3x^2 + 3x − 9))/color(blue)((x^4 - 81)#

We know the Difference of Squares Identity which says:
#color(blue)(a^2 - b^2 = (a+b)(a-b)#

So , #color(blue)((x^4 - 81) = (x^2 +9)(x^2-9)#

Here, again , #color(blue)((x^2-9) = (x+3)(x-3)#

Now, the expression becomes:
#color(red)((x^3 − 3x^2 + 3x − 9))/color(blue)((x^2+9)(x+3)(x-3)#

Factoring the numerator by grouping:
#color(red)((x^3 − 3x^2 + 3x − 9)) = x^2(x-3) + 3(x-3)#
# = color(red)( (x^2 +3)(x-3)#

The expression can now be written as:
# color(red)( (x^2 +3)cancel((x-3)))/ color(blue)((x^2+9)(x+3)cancel(x-3)#

# = color(red)( (x^2 +3)/ color(blue)((x^2+9)(x+3)#