Algebra and Functions: Partial Fractions (P4)
Edexcel International A-Level MathematicsΒ· 2018 Specification (Issue 3) WMA14 Β§2.1Β· 25 min read
1. Step 1: Identify and Rewrite Improper Rational Functionsβ β ββββ± 5 min
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Improper Rational Function
A rational function where (\deg(\text{numerator}) \geq \deg(\text{denominator})). Must be rewritten via polynomial division as a polynomial plus a proper rational function before partial fraction decomposition.
Example:
(\frac{2x^3 + 5x^2 - x + 1}{x^2 + 2x - 3}) is improper as (\deg(\text{numerator})=3 > \deg(\text{denominator})=2)
Rewrite (\frac{2x^3 + 5x^2 - x + 1}{x^2 + 2x - 3}) as a polynomial plus a proper rational function.
- 1
Confirm the function is improper: (\deg(\text{numerator})=3), (\deg(\text{denominator})=2), so polynomial division is required.
- 2
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The rewritten form is (2x + 1 + \frac{3x + 4}{x^2 + 2x - 3}), where (\frac{3x + 4}{x^2 + 2x - 3}) is a proper rational function.
Exam tip:
Always check the degree of numerator and denominator first: method marks are deducted for skipping polynomial division for improper fractions.
2. Partial Fractions: Distinct Linear Factors Denominatorβ β β βββ± 7 min
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Partial Fractions (Distinct Linear Factors)
Decomposition of a proper rational function into sum of fractions with constant numerators and unique linear denominators. Constants (A, B) are solved via substitution or equating coefficients.
Decompose (\frac{3x + 4}{(x - 1)(x + 3)}) into partial fractions.
- 1
Set up the decomposition form for distinct linear factors: (\frac{3x +4}{(x-1)(x+3)} = \frac{A}{x-1} + \frac{B}{x+3})
- 2
Multiply both sides by the denominator to eliminate fractions: (3x + 4 = A(x + 3) + B(x - 1))
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Substitute (x=1) (root of first factor): (3(1) + 4 = A(4) \implies 7 = 4A \implies A = \frac{7}{4})
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Substitute (x=-3) (root of second factor): (3(-3) + 4 = B(-4) \implies -5 = -4B \implies B = \frac{5}{4})
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Final decomposition: (\frac{7}{4(x-1)} + \frac{5}{4(x+3)})
3. Partial Fractions: Repeated Linear Factors Denominatorβ β β β ββ± 8 min
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Partial Fractions (Repeated Linear Factor)
Decomposition for denominators including a squared linear term: you need one fraction for each power of the repeated factor up to its exponent, not just the highest power.
Decompose (\frac{7x^2 + 19x + 16}{(x +1)(x + 2)^2}) into partial fractions.
- 1
Set up the decomposition form correctly, including terms for both the first and second power of the repeated factor: (\frac{7x^2 +19x +16}{(x+1)(x+2)^2} = \frac{A}{x+1} + \frac{B}{x+2} + \frac{C}{(x+2)^2})
- 2
Multiply through by the denominator: (7x^2 + 19x + 16 = A(x+2)^2 + B(x+1)(x+2) + C(x+1))
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Substitute (x=-1) (root of distinct factor): (7(1) + 19(-1) + 16 = A(1) \implies 4 = A)
- 4
Substitute (x=-2) (root of repeated factor): (7(4) + 19(-2) +16 = C(-1) \implies 6 = -C \implies C = -6)
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Equate coefficients of (x^2) to find (B): (7 = A + B \implies 7 = 4 + B \implies B = 3)
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Final decomposition: (\frac{4}{x+1} + \frac{3}{x+2} - \frac{6}{(x+2)^2})
Exam tip:
A very common mistake is omitting the linear power term for the repeated factor: always remember you need (n) terms for a factor raised to the (n)th power.
4. Exam Applications of Partial Fractionsβ β β βββ± 5 min
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Partial fractions are not tested in isolation in P4: they are a supporting technique for three core topics: binomial expansion of rational functions, integration of rational functions, and simplifying differentiation of complex rational functions to avoid repeated quotient rule use.
Show that (\int \frac{3x +4}{(x-1)(x+3)} dx = \frac{7}{4}\ln|x-1| + \frac{5}{4}\ln|x+3| + c)
- 1
First decompose the integrand into partial fractions (from earlier example): (\frac{3x+4}{(x-1)(x+3)} = \frac{7}{4(x-1)} + \frac{5}{4(x+3)})
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Integrate term by term using the standard rule (\int \frac{k}{ax + b} dx = \frac{k}{a}\ln|ax + b| + c)
- 3
5. Common Pitfalls
Wrong move:
Skipping polynomial division for improper rational functions before starting partial fraction decomposition.
Why:
Partial fraction decomposition only works for proper rational functions; you will get an incorrect result and lose method marks if you skip this step.
Correct move:
Always compare the degree of numerator and denominator first; if (\deg(\text{numerator}) \geq \deg(\text{denominator})), perform polynomial long division first to get a polynomial plus a proper rational function.
Wrong move:
Omitting the linear power term for a repeated linear factor, e.g. writing (\frac{A}{x+1} + \frac{B}{(x+2)^2}) for a denominator ((x+1)(x+2)^2).
Why:
The decomposition is incomplete, so the constants you calculate will not satisfy the original equation, leading to full loss of marks for the question.
Correct move:
Include one term for each power of the repeated factor up to its exponent: for ((cx + d)^2), include both (\frac{B}{cx +d}) and (\frac{C}{(cx +d)^2}).
Wrong move:
Forgetting to include the polynomial term from polynomial division in your final partial fraction decomposition.
Why:
Your final result will only represent the proper fraction part, not the original improper function, leading to incorrect answers in follow-up applications (integration, binomial expansion).
Correct move:
Add the polynomial obtained from division to your partial fraction decomposition of the proper remainder fraction to get the full decomposition of the original function.
Wrong move:
Using the same constant letter for multiple terms, or mixing up constants when substituting values.
Why:
This leads to arithmetic errors when solving for constants, and examiners will penalize incorrect constant values even if your set-up is correct.
Correct move:
Use distinct letters for each unknown constant, and double-check your substitution calculations before proceeding.
Wrong move:
Attempting to decompose fractions with irreducible quadratic denominators (e.g. (x^2 + 2)) for P4 questions.
Why:
Irreducible quadratic denominators are explicitly out of scope for Edexcel IAL P4; any such question would be for Further Maths, and attempting to decompose them will waste time and lead to errors.
Correct move:
Only process denominators that are products of linear factors, including at most one repeated linear factor, as per P4 specification.
6. Quick Reference Cheatsheet
Denominator Type | Decomposition Form | First Step Required? |
|---|---|---|
Distinct linear factors: ((ax + b)(cx + d)(ex + f)) | (\frac{A}{ax+b} + \frac{B}{cx+d} + \frac{C}{ex+f}) | No, if function is proper |
Repeated linear factor: ((ax + b)(cx + d)^2) | (\frac{A}{ax+b} + \frac{B}{cx+d} + \frac{C}{(cx+d)^2}) | No, if function is proper |
Improper rational function ((\deg(N) \geq \deg(D))) | Polynomial (Q(x)) + partial fractions of remainder fraction | Yes: perform polynomial division first to get (Q(x) + \frac{R(x)}{D(x)}) where (\deg(R) < \deg(D)) |
7. Frequently Asked
Do I need to learn partial fractions with quadratic denominators for P4?
No, irreducible quadratic denominators are explicitly excluded from Edexcel IAL P4. You only need to handle denominators that are products of distinct linear factors or products of linear factors including one repeated linear term.
When do I need to do polynomial division before partial fraction decomposition?
You must perform polynomial division first if your rational function is improper, i.e. the degree of the numerator is equal to or higher than the degree of the denominator.
Going deeper
What's Next
Now that you have mastered partial fraction decomposition for Edexcel IAL P4, you can apply this technique to the three core topics it supports in the unit. Partial fractions are a prerequisite for binomial expansion of rational functions, where they allow you to break complex fractions into simpler forms that fit the standard binomial expansion formula for negative and fractional exponents. They are also essential for integrating rational functions, where each partial fraction term can be integrated directly using the natural logarithm rule, and can simplify differentiation tasks by avoiding the need for repeated quotient rule applications. Make sure you practice combining partial fractions with these follow-up topics to score full marks on P4 exam questions.