Non-Repeated Linear Factors

When the denominator of a proper fraction is a product of distinct linear factors, $g(x)=(a_1x+b_1)\cdots(a_nx+b_n)$, assign one unknown constant per factor:

Multiplying both sides by $g(x)$ clears every denominator and leaves a polynomial identity that must hold for all $x$. The quickest way to find each $A_k$ exploits this: substituting $x=-b_k/a_k$, the root of the $k$-th factor, makes every term on the right except the one containing $A_k$ vanish (since each of the other factors is still present as a multiplier), isolating $A_k$ in one step. This is often called the cover-up rule, since it amounts to covering up the factor $(a_kx+b_k)$ in the original fraction and evaluating what remains at $x=-b_k/a_k$. …