How do you simplify (3z-6)/(3z^2-12)3z−63z2−12?
1 Answer
Explanation:
The first step in simplifying is to factorise the numerator/denominator.
Numerator:
3z - 6 has a
color(blue)"common factor"common factor 0f 3.
rArr3z-6=3(z-2)larr" factorised form"⇒3z−6=3(z−2)← factorised form Denominator:
3z^2-12" also has a common factor of 3"3z2−12 also has a common factor of 3
rArr3z^2-12=3(z^2-4)⇒3z2−12=3(z2−4) The factor
z^2-4" is a difference of squares"z2−4 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)
