Algebraic Methods Pure Mathematics 3 (A Level)

1. Partial Fractions

Partial fractions are used to break down rational expressions into simpler components.

1.1 Proper and Improper Fractions

A rational expression

$\frac{\mathrm{P(x)}}{\mathrm{Q(x)}}$

is proper if:

• degree of P(x) < degree of Q(x)

If degree of P(x) ≥ degree of Q(x), it is improper and must be simplified using algebraic division.

1.2 Distinct Linear Factors

If the denominator contains linear factors such as (x − a)(x − b):

$\frac{A}{x-a}+\frac{B}{x-b}$

Example:
Resolve: $\frac{5}{x+1}$ over (x − 1)(x + 2).

1.3 Repeated Linear Factors

If the denominator contains (x − a)²:

$\frac{A}{x-a}+\frac{B}{(x-a)2^{}}$

1.4 Quadratic Factors

If the denominator has an irreducible quadratic:

$\frac{\mathrm{Ax}+B}{x{2}^{}+\mathrm{px}+q}$

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2. Algebraic Division

Used to convert improper rational expressions into proper ones.

Divide: $\frac{2x{2}^{}+3x+1}{x+1}$

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3. Binomial Expansion

3.1 Positive Integer Powers

${(a+b)}^n$

General term:

$T_{r+1}=C{n}_r{a}^{n-r}{b}^r$

3.2 Fractional and Negative Powers

${(1+x)}^n$

Valid for |x| < 1

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4. Proof by Induction

1. Prove true for n = 1
2. Assume true for n = k
3. Show true for n = k + 1
4. Conclude true for all n ≥ 1

Prove: $\sum _{i=1}^{n}i=\frac{n(n+1)}{2}$

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5. Factor & Remainder Theorem

$f\left(a\right)=0$

⇒ (x − a) is a factor of f(x)

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6. Exam Tips

• Always write the correct partial fraction form first
• State the binomial general term clearly
• Show all steps in induction proofs
• Use algebraic division for improper fractions

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7. Practice Questions

1. Resolve 7/(x² − 1) into partial fractions
2. Expand (1 − 3x)⁻² up to x²
3. Prove by induction: 1² + 2² + … + n²