Equivalent Representations and the Binomial Theorem

Each form answers a different question

The same rational function can be written three ways, and AP questions are engineered so that only one form makes the answer visible:

• Factored form $\dfrac{a(x-z_1)(x-z_2)}{(x-v_1)(x-v_2)}$ — exposes zeros, vertical asymptotes and holes.
• Standard form $\dfrac{a_nx^n+\cdots}{b_mx^m+\cdots}$ — exposes end behavior via leading terms.
• Quotient–remainder form $q(x)+\dfrac{r(x)}{d(x)}$, from polynomial long division with $\deg r<\deg d$ — exposes the slant (or polynomial) asymptote $y=q(x)$, because the remainder term vanishes at infinity.

Long division is the bridge between forms, and the Binomial Theorem is the fast lane for expanding powers:

The coefficients $\binom{n}{k}$ are row $n$ of Pascal's triangle. Watch the sign: in $(a-b)^n$, the term with $b^k$ carries $(-1)^k$, and any coefficient inside $b$ gets raised to the $k$-th power too.