Question

How do you simplify `(x^3 - 9x) / (x^2 - 7x + 12)`?

Answer 1

`(x(x+3))/(x-4)`

Explanation:

`"factorise numerator and denominator"`

`color(magenta)"factor numerator"`

`"take out a "color(blue)"common factor "x`

`=x(x^2-9)`

`x^2-9" is a "color(blue)"difference of squares"`

`"which factors in general as"`

`•color(white)(x)a^2-b^2=(a-b)(a+b)`

`"here "a=x" and "b=3`

`rArrx^2-9=(x-3)(x+3)`

`rArrx^3-9x=x(x-3)(x+3)larrcolor(red)"factorised form"`

`color(magenta)"factor denominator"`

`"the factors of + 12 which sum to - 7 are - 3 and - 4"`

`rArrx^2-7x+12=(x-3)(x-4)larrcolor(red)"factored form"`

`rArr(x^3-9x)/(x^2-7x+12)`

`=(x(x-3)(x+3))/((x-3)(x-4))`

`"cancel the "color(blue)"common factor "(x-3)`

`=(xcancel((x-3))(x+3))/(cancel((x-3))(x-4))=(x(x+3))/(x-4)`

`"with restriction "x!=4`

Answer 2

`(x(x+3))/(x-4`

Explanation:

`(x^3-9x)/(x^2-7x+12)`
`color(teal)(=(x(x^2-9))/(x^2-3x-4x+12)`

`color(blue)(=(x(x+3)(x-3))/((x-3)(x-4))`

`color(magenta)(=(x(x+3)cancel((x-3)))/(cancel((x-3))(x-4))`

`color(green)(=(x(x+3))/(x-4`

Ans that's your answer.

P.S.: Isn't the solution `color(blue)c`olorful?