Mathematics

This is partial fraction decomposition with a repeated linear factor. The key insight is that when $(x-3)$ appears squared in the denominator, you need separate terms for both $(x-3)$ and $(x-3)^2$—each captures a different "layer" of the denominator's structure. Multiply both sides by the common denominator $x(x-3)^2$ to clear fractions, then match coefficients or substitute strategic values of $x$ to find $A$, $B$, and $C$.

This is partial fraction decomposition with a repeated linear factor. The key insight is that when $(x-3)$ appears squared in the denominator, you need separate terms for both $(x-3)$ and $(x-3)^2$—each captures a different "layer" of the denominator's structure. Multiply both sides by the common denominator $x(x-3)^2$ to clear fractions, then match coefficients or substitute strategic values of $x$ to find $A$, $B$, and $C$.