Question

How do you simplify `(x-3)/(x-5) div (x^3-27)/(x^3+8)*(x^2-25)/(x + 5)`?

Answer 1

`((x+2)(x^2-2x+4))/(x^2+3x+9)`

Explanation:

Recall the following:

`1`. Difference of cubes: `color(orange)((a^3-b^3)=(a-b)(a^2+ab+b^2))`

`2`. Sum of cubes: `color(blue)((a^3+b^3)=(a+b)(a^2-ab+b^2))`

`3`. Difference of squares: `color(purple)((a^2-b^2)=(a+b)(a-b))`

The Simplification Process
`1`. Start by factoring the polynomials in the numerator and denominator of the second fraction.

`(x-3)/(x-5)-:(x^3-27)/(x^3+8)*(x^2-25)/(x+5)`

`=(x-3)/(x-5)-:(color(orange)((x-3)((x)^2+(3)(x)+(3)^2)))/(color(blue)((x+2)((x)^2-(2)(x)+(2)^2)))*(x^2-25)/(x+5)`

`2`. Simplify.

`=(x-3)/(x-5)-:(color(orange)((x-3)(x^2+3x+9)))/(color(blue)((x+2)(x^2-2x+4)))*(x^2-25)/(x+5)`

`3`. Factor the polynomials in the numerator and denominator of the third fraction.

`=(x-3)/(x-5)-:(color(orange)((x-3)(x^2+3x+9)))/(color(blue)((x+2)(x^2-2x+4)))*(color(purple)((x+5)(x-5)))/(x+5)`

`4`. To divide the first fraction by the second fraction, rewrite the second fraction as its reciprocal.

`=(x-3)/(x-5)*(color(blue)((x+2)(x^2-2x+4)))/(color(orange)((x-3)(x^2+3x+9)))*color(purple)((x+5)(x-5))/(x+5)`

`5`. Cancel out any factors which appear in the numerator and denominator as a pair.

`=color(red)cancelcolor(black)((x-3))/color(teal)cancelcolor(black)((x-5))*(color(blue)((x+2)(x^2-2x+4)))/(color(orange)(color(red)cancelcolor(orange)((x-3))(x^2+3x+9)))*color(purple)(color(magenta)cancelcolor(purple)((x+5))color(teal)cancelcolor(purple)((x-5)))/color(magenta)cancelcolor(black)((x+5))`

`6`. Rewrite the expression.

`=color(green)(|bar(ul(color(white)(a/a)((x+2)(x^2-2x+4))/(x^2+3x+9)color(white)(a/a)|)))`