Partial Fractions as an Algebraic Identity

Definition A rational expression $\frac{P(x)}{D(x)}$, with $P$ and $D$ polynomials and $D$ not the zero polynomial, is proper when $P$ is zero or the degree of $P$ is less than the degree of $D$. Partial fraction decomposition applies only to proper expressions. If the expression is improper, first divide to get $P = DQ + R$ with $R$ zero or of lower degree than $D$, and then decompose the proper part $\frac{R}{D}$.

Definition To add rational expressions, rewrite each over a common denominator, usually the product of the distinct factors to their highest powers, and add the numerators. For nonzero polynomials $B$ and $D$,

valid wherever $B$ and $D$ are both nonzero. Recombining over a common denominator is how every partial fraction result is verified.

Definition A linear factor of the denominator is a factor $ax + b$ with real $a \ne 0$. In a decomposition, a linear factor that appears once contributes one term $\frac{A}{ax + b}$ with a constant numerator $A$. A linear factor that appears to the power $k$ contributes $k$ terms, one for each power,

Definition A quadratic $ax^2 + bx + c$ with real coefficients and $a \ne 0$ is irreducible over the reals when it has no real roots, that is when $b^2 - 4ac < 0$. Such a factor cannot be split into real linear factors. In a decomposition, an irreducible quadratic that appears once contributes one term with a numerator of degree at most one,

A repeated irreducible quadratic contributes one such term for each power, as with linear factors.

Definition The numerators $A$, $B$, $C$ and so on in the chosen form are unknown coefficients. Multiplying both sides by the original denominator gives an equation between two polynomials. Coefficient matching sets the coefficients of each power of $x$ equal on both sides, which gives a system of linear equations for the unknowns. Substituting convenient values of $x$, such as the roots of linear factors, gives equations for the same unknowns and is often faster.

Definition Let $\frac{P(x)}{D(x)}$ be a proper rational expression with real coefficients, and factor $D$ into real linear factors and irreducible quadratic factors. Its partial fraction decomposition writes it as a sum of terms, one term $\frac{A}{(ax + b)^j}$ with a real constant $A$ for each power $j$ up to the multiplicity of each linear factor and one term $\frac{Bx + C}{(ax^2 + bx + c)^j}$ with real constants $B$ and $C$ for each power $j$ up to the multiplicity of each irreducible quadratic factor. Each term has its own unknown constants. Such a decomposition always exists and its coefficients are unique. Each term is called a partial fraction.

Definition A partial fraction decomposition is an equality of fractions that holds exactly where the original denominator $D(x)$ is nonzero. Multiplying both sides by $D(x)$ gives an equation between two polynomials that is true at every real $x$ except the finitely many roots of $D$. The difference of the two sides is then a polynomial with infinitely many roots, and the only such polynomial is the zero polynomial. So the cleared equation is a polynomial identity, true for every real $x$, roots of $D$ included. That is why substituting a root of the denominator into the cleared equation is legitimate, even though the fractions themselves are undefined there.