In VCE Specialist Mathematics, partial fraction decomposition is the process of expressing a complex rational function as a sum of simpler fractions. This technique is essential for sketching complex rational graphs and is a prerequisite skill for advanced integration techniques.
A rational function is defined as the quotient of two polynomials:
\$\(f(x) = \frac{g(x)}{h(x)}\)\$
where \(g(x)\) and \(h(x)\) are polynomials.
Before decomposing into partial fractions, you must determine the type of fraction:
* Proper Fraction: The degree of the numerator \(g(x)\) is less than the degree of the denominator \(h(x)\).
* Improper Fraction: The degree of the numerator \(g(x)\) is greater than or equal to the degree of the denominator \(h(x)\).
VCAA FOCUS: If a fraction is improper, you must perform polynomial long division first. The function is rewritten in the form:
\$\(\frac{g(x)}{h(x)} = q(x) + \frac{r(x)}{h(x)}\)\$
where \(q(x)\) is the quotient and \(r(x)\) is the remainder (which is now a proper fraction).
The form of the partial fractions depends entirely on the factors of the denominator \(h(x)\).
For every distinct linear factor \((ax + b)\) in the denominator, there is a partial fraction of the form:
\$\(\frac{A}{ax + b}\)\$
Example:
\$\(\frac{4x + 2}{(x - 1)(x + 1)} = \frac{A}{x - 1} + \frac{B}{x + 1}\)\$
For every repeated linear factor \((ax + b)^n\) in the denominator, there must be \(n\) partial fractions with increasing powers in the denominator:
\$\(\frac{A}{ax + b} + \frac{B}{(ax + b)^2} + \dots + \frac{N}{(ax + b)^n}\)\$
Example:
\$\(\frac{3x - 4}{(2x - 3)(x + 5)^2} = \frac{A}{2x - 3} + \frac{B}{x + 5} + \frac{C}{(x + 5)^2}\)\$
COMMON MISTAKE: Students often forget the first-degree term for repeated factors. For \((x+5)^2\), you must include both the \(\frac{B}{x+5}\) and the \(\frac{C}{(x+5)^2}\) terms.
A quadratic factor \(ax^2 + bx + c\) is irreducible if it cannot be factored over the real numbers (\(\mathbb{R}\)). This occurs when the discriminant is negative (\(\Delta = b^2 - 4ac < 0\)).
For every irreducible quadratic factor, the numerator of the partial fraction must be a linear expression:
\$\(\frac{Ax + B}{ax^2 + bx + c}\)\$
Example:
\$\(\frac{3x - 4}{(2x - 3)(x^2 + 5)} = \frac{A}{2x - 3} + \frac{Bx + C}{x^2 + 5}\)\$
Similar to repeated linear factors, these require terms with increasing powers, each with a linear numerator:
\$\(\frac{Ax + B}{ax^2 + bx + c} + \frac{Cx + D}{(ax^2 + bx + c)^2}\)\$
To resolve a fraction into partial fractions, follow these three steps:
EXAM TIP: Use a combination of both methods! Use substitution to find constants associated with linear factors quickly, then equate the highest power coefficient or the constant term to find the remaining variables.
| Denominator Factor Type | Factor in Denominator | Partial Fraction Contribution |
|---|---|---|
| Linear | \((ax + b)\) | \(\frac{A}{ax + b}\) |
| Repeated Linear | \((ax + b)^2\) | \(\frac{A}{ax + b} + \frac{B}{(ax + b)^2}\) |
| Irreducible Quadratic | \((ax^2 + bx + c)\) | \(\frac{Ax + B}{ax^2 + bx + c}\) |
| Repeated Quadratic | \((ax^2 + bx + c)^2\) | \(\frac{Ax + B}{ax^2 + bx + c} + \frac{Cx + D}{(ax^2 + bx + c)^2}\) |
KEY TAKEAWAY: The degree of the numerator in your partial fraction setup must always be exactly one less than the degree of the base factor in the denominator (e.g., a degree 2 quadratic base requires a degree 1 linear numerator).
Partial fraction decomposition is primarily used in two areas of the VCE course:
1. Graphing Rational Functions: Decomposing a function helps identify vertical asymptotes and the behavior of the graph near those asymptotes.
2. Integration (Unit 4): It is impossible to integrate \(\int \frac{1}{x^2-1} dx\) directly, but it is simple to integrate its partial fractions: \(\int (\frac{1/2}{x-1} - \frac{1/2}{x+1}) dx\).
STUDY HINT: Practice identifying the form of the decomposition before worrying about the algebra of solving for \(A, B,\) and \(C\). Most errors occur in the initial setup or during the polynomial long division of improper fractions.
