Answer:
[tex]\boxed{(\frac{f}{g})(x)=\frac{x^2+3x+9}{3}}[/tex]
Step-by-step explanation:
The given functions are
[tex]f(x)=x^3-27[/tex]
and
[tex]g(x)=3x-9[/tex]
We want to find [tex](\frac{f}{g})(x)[/tex] of the given functions.
[tex](\frac{f}{g})(x)=\frac{f(x)}{g(x)}[/tex]
[tex]\Rightarrow (\frac{f}{g})(x)=\frac{x^3-27}{3x-9}[/tex]
[tex]\Rightarrow (\frac{f}{g})(x)=\frac{x^3-3^3}{3x-9}[/tex]
Recall that;
[tex]a^3-b^3=(a-b)(a^2+ab+b^2)[/tex]
We apply this property to factor the numerator to obtain;
[tex](\frac{f}{g})(x)=\frac{(x-3)(x^2+3x+9)}{3(x-3)}[/tex]
We cancel out the common factors to obtain;
[tex](\frac{f}{g})(x)=\frac{x^2+3x+9}{3}[/tex]
