Polynomials & Partial Fractions
Polynomial Division
$P(x)=\text{divisor}\times Q(x)+R(x)$
Remainder Theorem
If $P(x)$ is divided by $x-c$, remainder is $\text{f}(c)$.
If $P(x)$ is divided by $ax-b$, remainder is $\text{f}\left(\dfrac{b}{a}\right)$.
If $P(x)$ is divided by $ax-b$, remainder is $\text{f}\left(\dfrac{b}{a}\right)$.
Factor Theorem
If $x-c$ is a factor of $P(x)$, $\text{f}(c)=0$.
If $ax+b$ is a factor of $P(x)$, $\text{f}\left(-\dfrac{b}{a}\right)=0$.
If $ax+b$ is a factor of $P(x)$, $\text{f}\left(-\dfrac{b}{a}\right)=0$.
Cubic Polynomials
$a^3+b^3 = (a+b)(a^2-ab+b^2)$
$a^3-b^3 = (a-b)(a^2+ab+b^2)$
$a^3-b^3 = (a-b)(a^2+ab+b^2)$
Partial Fractions
$1.\,\dfrac{f(x)}{(ax + b)(cx+d)} = \dfrac{A}{ax+b} + \dfrac{B}{cx+d}$
$2.\,\dfrac{f(x)}{(ax + b)(cx+d)^2} = \dfrac{A}{ax+b} + \dfrac{B}{cx+d} + \dfrac{C}{({cx+d})^2}$
$3.\,\dfrac{f(x)}{(ax + b)(x^2+c)} = \dfrac{A}{ax+b} + \dfrac{Bx+C}{x^2+c}$
*$\dfrac{f(x)}{(ax + b)(x^2)} = \dfrac{A}{ax+b} + \dfrac{B}{x}+\dfrac{C}{x^2}$
$2.\,\dfrac{f(x)}{(ax + b)(cx+d)^2} = \dfrac{A}{ax+b} + \dfrac{B}{cx+d} + \dfrac{C}{({cx+d})^2}$
$3.\,\dfrac{f(x)}{(ax + b)(x^2+c)} = \dfrac{A}{ax+b} + \dfrac{Bx+C}{x^2+c}$
*$\dfrac{f(x)}{(ax + b)(x^2)} = \dfrac{A}{ax+b} + \dfrac{B}{x}+\dfrac{C}{x^2}$
