Question

How do you simplify `(3z-6)/(3z^2-12)`?

Answer 1

`1/(z+2)`

Explanation:

The first step in simplifying is to factorise the numerator/denominator.

Numerator:

3z - 6 has a `color(blue)"common factor"` 0f 3.

`rArr3z-6=3(z-2)larr" factorised form"`

Denominator:

`3z^2-12" also has a common factor of 3"`

`rArr3z^2-12=3(z^2-4)`

The factor `z^2-4" is a difference of squares"` and, in general is factorised as follows.

`color(red)(bar(ul(|color(white)(a/a)color(black)(a^2-b^2=(a-b)(a+b))color(white)(a/a)|)))`

now `(z)^2=z^2" and " (2)^2=4rArra=z" and " b=4`

Thus `z^2-4=(z-2)(z+2)`

`rArr3z^2-12=3(z-2)(z+2)larr" in factorised form"`

Transferring these results into the original expression.

`rArr(3(z-2))/(3(z-2)(z+2))`

cancelling common factors on numerator/denominator gives.

`(cancel(3)^1cancel((z-2)))/(cancel(3)^1cancel((z-2))(z+2))=1/(z+2)`