Partial Fractions Decomposition Reference Cheat Sheet

Key Facts

• A rational expression $\frac{P(x)}{Q(x)}$ is proper when $\deg P(x)<\deg Q(x)$, and partial fractions should be set up only after this is true.
• For a distinct linear factor $x-a$, use the form $\frac{A}{x-a}$.
• For repeated linear factors $(x-a)^n$, use the full chain $\frac{A_1}{x-a}+\frac{A_2}{(x-a)^2}+\cdots+\frac{A_n}{(x-a)^n}$.
• For an irreducible quadratic factor $ax^2+bx+c$, use the form $\frac{Ax+B}{ax^2+bx+c}$.
• For repeated irreducible quadratic factors $(ax^2+bx+c)^n$, include $\frac{A_1x+B_1}{ax^2+bx+c}+\frac{A_2x+B_2}{(ax^2+bx+c)^2}+\cdots+\frac{A_nx+B_n}{(ax^2+bx+c)^n}$.
• After writing the partial fraction form, multiply both sides by the least common denominator to create a polynomial identity.
• Coefficients can be found by substituting convenient values of $x$ or by matching coefficients of equal powers of $x$.
• If $\deg P(x)\ge \deg Q(x)$, use polynomial division first: $\frac{P(x)}{Q(x)}=S(x)+\frac{R(x)}{Q(x)}$ with $\deg R(x)<\deg Q(x)$.

Vocabulary

Rational expression

A quotient of two polynomials, written as $\frac{P(x)}{Q(x)}$ where $Q(x)\ne 0$.

Proper rational expression

A rational expression in which the numerator has smaller degree than the denominator, so $\deg P(x)<\deg Q(x)$.

Linear factor

A first-degree factor of a polynomial, such as $x-a$ or $mx+b$.

Irreducible quadratic

A quadratic factor $ax^2+bx+c$ that cannot be factored into real linear factors.

Repeated factor

A factor that appears with an exponent greater than $1$, such as $(x-3)^2$.

Polynomial identity

An equation between polynomials that is true for all allowed values of $x$.