Polynomials & Partial Fractions

Polynomial Division

$P(x)=\text{divisor}\times Q(x)+R(x)$

If $P(x)$ is divided by $x-c$, remainder is $f(c)$.

If $P(x)$ is divided by $ax-b$, remainder is $f\left(\dfrac{b}{a}\right)$.

If $x-c$ is a factor of $P(x)$, $f(c)=0$.

If $ax+b$ is a factor of $P(x)$, $f\left(-\dfrac{b}{a}\right)=0$.

Sum of Cubes

$a^3+b^3 = (a+b)(a^2-ab+b^2)$

Difference of Cubes

$a^3-b^3 = (a-b)(a^2+ab+b^2)$

Distinct Linear Factors

$\dfrac{f(x)}{(ax + b)(cx+d)} = \dfrac{A}{ax+b} + \dfrac{B}{cx+d}$

Repeated Linear Factor

$\dfrac{f(x)}{(ax + b)(cx+d)^2} = \dfrac{A}{ax+b} + \dfrac{B}{cx+d} + \dfrac{C}{(cx+d)^2}$

Quadratic Factor

$\dfrac{f(x)}{(ax + b)(x^2+c)} = \dfrac{A}{ax+b} + \dfrac{Bx+C}{x^2+c}$

*Special case

$\dfrac{f(x)}{(ax + b)(x^2)} = \dfrac{A}{ax+b} + \dfrac{B}{x}+\dfrac{C}{x^2}$