Question

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

Answer 1

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

Explanation:

`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)`