Concepts

transformation

first stretch (distinguish between vertical and horizontal stretch) then apply shift

stretch first, then shift

extended binomial expansion

for \((a+b)^k, \\ k \in R\) :expansion:

\[ (a+b)^k =a^k(1+\frac{b}{a})^k=a^k \sum_{n=0}^{\infty} \binom{k}{n} (\frac{b}{a})^n \] to converge, satisfy: \(|\frac{b}{a}|<1\)

Polynomial Division

for two polynomial \(g(x)\) and \(f(x)\), we have \(\frac{g(x)}{f(x)}\). If degree of \(g(x)\) is bigger than \(f(x)\), use polynomial division

Partial Fraction

for two polynomial \(g(x)\) and \(f(x)\), we have \(\frac{g(x)}{f(x)}\). If degree of \(g(x)\) is smaller than \(f(x)\), use Partial Fraction

ensure \(f(x)\) is fully factorized

for \(f(x)\), if it has form of: 1. \((x+a_0)\): \(\frac{A}{x+a_0}\) 2. \((x+a_0)^n\): \(\frac{A_1}{(x+a_0)}+\frac{A_2}{(x+a_0)^2}+...+\frac{A_n}{(x+a_0)^n}\) 3. \((ax^2+bx+c)\): use factor of \(\frac{Ax+B}{(ax^2+bx+c)}\)

常见敌人

quadratic function’s root and coefficients: remember always use \(\Delta\) to test if the roots exist