Question

How do you divide and simplify `\frac { x - 9} { x ^ { 2} - 81} \div \frac { x ^ { 2} - 7x - 30} { x ^ { 2} + 12x + 27}`?

Answer 1

`=1/(x-10)`

Explanation:

step 1

Use the rule for division of fractions: invert the second fraction and change to multiply

`(x-9)/(x^2-81)xx(x^2+12x+27)/(x"-7x-30)`

step 2

factorise as much as possible, cancelling as we go along

a) difference of squares in denominator in the first fraction

`=cancel(color(red)((x-9)))/((x+9)cancel(color(red)((x-9))))xx(x^2+12x+27)/(x^2-7x-30)`

`1/((x+9))xx(x^2+12x+27)/(x^2-7x-30)`

b) quadratic factorisation in the second fraction

`1/((x+9))xx((x+9)(x+3))/((x-10)(x+3))`

c) Further cancelling

`1/(cancelcolor(red)((x+9)))xx(cancelcolor(red)((x+9))cancel(color(blue)((x+3))))/((x-10)cancel(color(blue)((x+3)))`

`=1/(x-10)`