Partial Fraction Decomposition (Quadratic Factors)

Definition A rational function $\frac{P(x)}{Q(x)}$, a quotient of polynomials with $Q$ not the zero polynomial, is proper if $\deg P < \deg Q$. Only proper rational functions are decomposed. An improper one is first divided, which gives $\frac{P}{Q} = S + \frac{R}{Q}$ with a polynomial $S$ and either $R = 0$ or $\deg R < \deg Q$, valid wherever $Q(x) \ne 0$. A proper fraction over a quadratic has a numerator of degree at most $1$, which is where linear numerators come from.

Definition A quadratic $ax^2 + bx + c$ with real coefficients and $a \ne 0$ is irreducible over the real numbers if it cannot be written as a product of two linear polynomials with real coefficients. This happens exactly when it has no real zero, that is when its discriminant is negative,

Definition Every polynomial with real coefficients and degree at least $1$ can be written as its leading coefficient times a product of linear factors $x - r$ with real $r$ and irreducible quadratic factors $x^2 + px + q$ with $p^2 - 4q < 0$. This factorization is unique apart from the order of the factors. The reason comes from the fundamental theorem of algebra. The nonreal zeros of a real polynomial come in conjugate pairs $a + bi$ and $a - bi$ with $b \ne 0$, and each pair combines into a real quadratic factor

whose discriminant $-4b^2$ is negative. So the only irreducible real polynomials are the linear ones and the quadratics with negative discriminant.

Definition Let $\frac{P}{Q}$ be a proper rational function with real coefficients, and factor $Q$ over the reals. The decomposition contains the terms for the linear factors, as in the previous lesson, and each irreducible quadratic factor $ax^2 + bx + c$ that appears exactly once contributes one term

with real constants $B$ and $C$. Either constant may turn out to be zero, but the form must allow both.

Definition If an irreducible quadratic factor $ax^2 + bx + c$ appears exactly $m$ times in the real factorization of $Q$, it contributes $m$ terms, each with its own linear numerator,

With these terms and the terms for the linear factors and their powers, every proper rational function with real coefficients has exactly one decomposition, valid wherever $Q(x) \ne 0$.

Definition If a decomposition holds, then after multiplying both sides by $Q(x)$ they are polynomials that agree at all but finitely many $x$, so they are the same polynomial and the coefficients of each power of $x$ match. This gives a linear system with exactly one solution. Real zeros of $Q$ still give quick equations by substitution, but an irreducible quadratic factor has no real zero to substitute. Its constants come from matching coefficients, or from substituting other convenient values such as $x = 0$.