Solving Coefficients For Irreducible Quadratics

From a Partial Fraction Form to Its Coefficients

A partial fraction template specifies the shape of the terms in a decomposition, but it does not specify their coefficients. For an irreducible quadratic factor, the numerator must be linear: it has the form \(Ax+B\), with constants \(A\) and \(B\) to be determined.

The practical task is to turn the proposed decomposition into equations for those unknown constants. The reliable method is to multiply through by the common denominator. This produces a polynomial identity, which can be solved by matching coefficients or by evaluating it at judiciously selected values.

Because an irreducible quadratic has no real zeros, there is no real value of \(x\) to substitute that makes its factor vanish. This is why the cover-up method for distinct real linear factors does not directly give the coefficients here. Clearing denominators works without needing any real roots.

Why Clearing Denominators and Matching Coefficients Works

Suppose a rational function has already been written in the appropriate form, with distinct irreducible quadratic factors \(Q_1(x),\ldots,Q_m(x)\). Its unknown coefficients occur in numerators such as \(A_i x+B_i\). On any \(x\) where the denominators are nonzero, the proposed decomposition is an equality of rational expressions. Multiplying both sides by \(Q_1(x)\cdots Q_m(x)\) gives a polynomial identity.

After expanding, collect terms by powers of \(x\). Two polynomials are equal precisely when their corresponding coefficients are equal. In particular, if their difference is the zero polynomial, differentiating that difference \(j\) times and setting \(x=0\) gives \(j!\) times its coefficient of \(x^j\), which must be zero. Thus every coefficient in the difference is zero.

For two distinct quadratic factors, the standard form is \(\frac{Ax+B}{Q_1(x)}+\frac{Cx+D}{Q_2(x)}\). Multiplying through by \(Q_1(x)Q_2(x)\) gives \(P(x)=(Ax+B)Q_2(x)+(Cx+D)Q_1(x)\), where \(P(x)\) is the numerator of the original proper rational function. Expanding this equality creates a linear system in \(A,B,C,D\).

Worked Example: Matching Coefficients for Two Quadratics

Using Selected Substitutions

Coefficient matching is systematic, but expanding every coefficient equation is not always the shortest route. After clearing denominators, one may instead evaluate the resulting polynomial identity at selected values of \(x\). Each value gives an equation in the unknown coefficients. Values such as \(0,1,-1\) can be useful because signs and powers simplify.

These substitutions do not need to make a quadratic factor zero. In fact, an irreducible quadratic has no real zero to use. A set of evaluations is useful when it produces enough independent equations to determine the unknowns. If the cleared identity has degree at most \(n\), agreement at \(n+1\) distinct values is enough to establish equality of the two polynomials: their difference has degree at most \(n\), and a nonzero polynomial of degree at most \(n\) cannot have more than \(n\) distinct roots.

Worked Example: Solving by Selected Values

Separating Even and Odd Parts

When the quadratic factors involve \(x^2\) alone, it can be efficient to write \(y=x^2\) after clearing denominators. Terms involving \(x\) form the odd part of the identity, while terms without a factor of \(x\) form the even part. Each part can then be solved separately. This is still coefficient solving: the substitution reorganizes the equations rather than changing the required partial fraction form.

Worked Example: Three Quadratics and a Reduced System

A Common Expansion Error

A frequent mistake is to multiply a coefficient by the wrong power of \(x\). In the first example, an incorrect expansion might claim

The term \(4Ax\) comes from multiplying \(Ax\) by \(4\), not \(4Ax^2\). The correct expansion is

Using the incorrect version would assign the wrong powers to terms and therefore produce false coefficient equations. A dependable check is to distribute each product in two separate lines before collecting like powers. After solving, recombine the numerators over the common denominator; this catches errors in both expansion and algebra.

Choosing a Method and Checking the Result

Coefficient matching is dependable for any of these systems and is often clearest when there are only a few unknowns. Selected substitutions can reduce arithmetic when the cleared identity has a manageable degree and chosen values simplify the factors. A hybrid approach is also useful: expand enough to understand the structure, then use evaluations to obtain simpler equations.

In every case, substitutions are applied to the cleared polynomial identity, not used as a substitute for clearing denominators. Once the coefficients are found, recombine the proposed partial fractions. The resulting numerator must equal the original numerator, while the denominator remains the common product of the quadratic factors.