Algebra and Functions (A2 Unit 3: Pure Mathematics B) — WJEC A-Level Mathematics
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Algebra and Functions (A2 Unit 3: Pure Mathematics B) explained
Decomposing proper algebraic fractions into simpler partial fractions facilitates integration and binomial expansion.
Read the full explanation
For distinct linear factors, P(x)/((ax + b)(cx + d)) decomposes into A/(ax + b) + B/(cx + d). For repeated linear factors, P(x)/((ax + b)(cx + d)²) requires an escalating denominator: A/(ax + b) + B/(cx + d) + C/((cx + d)²). Multiplying through by the common denominator yields a polynomial identity. The unknown constants A, B, and C can be evaluated by substituting strategic roots (values of x that make individual linear factors zero) or by expanding and equating coefficients of corresponding powers of x. If the original fraction is improper, polynomial division must precede decomposition.
Your focus
- Decompose rational expressions with distinct linear factors into partial fractions.
- Decompose rational expressions containing repeated linear factors into partial fractions.
- Determine partial fraction constants using strategic substitution and equating coefficients.
Algebra and Functions (A2 Unit 3: Pure Mathematics B) exam tips
Marking Points
- stating the correct partial fraction template including repeated factor terms
- multiplying through to set up the identity and substituting values of x or equating coefficients
- solving accurately for constants A, B, and C and writing the final partial fraction decomposition
Examiner Tips
- 💡Use strategic substitution of x-values that make individual brackets zero to find constants quickly.
- 💡Check your decomposition by choosing a simple test value like x = 0 and evaluating both sides.
Common Mistakes
- omitting the single linear term B/(cx + d) when decomposing repeated factors ((cx + d)²)
- failing to perform polynomial long division first when given an improper rational fraction