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Topic 4 of 8 · Form 6

Pure 3: algebra, logarithms and numerical methods

Modulus equations and inequalities (e.g. |x − 2| < |2x + 1|, squaring both sides); polynomial division; factor and remainder theorems; partial fractions (linear, repeated and quadratic factors); binomial expansion for any rational n with validity.

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  1. PDFPure 3: algebra, logarithms and numerical methods – Mathematics notes (Cambridge A Level)46.9 KB

Algebra

Modulus equations and inequalities (e.g. |x − 2| < |2x + 1|, squaring both sides); polynomial division; factor and remainder theorems; partial fractions (linear, repeated and quadratic factors); binomial expansion for any rational n with validity.

Worked example: Solve |x − 2| < |2x + 1|.

(x − 2)² < (2x + 1)² → 3x² + 8x − 3 > 0 → (3x − 1)(x + 3) > 0 → x < −3 or x > ⅓

Logarithms and exponentials

Laws of logs, ln and eˣ; solving equations such as 3^(2x) − 4(3ˣ) + 3 = 0; using logarithms to linearise y = kxⁿ and y = kaˣ.

Numerical solution of equations

Sign-change methods; iteration xₙ₊₁ = F(xₙ) and convergence; giving roots to a stated accuracy.

Worked example: Use xₙ₊₁ = ∛(5 − xₙ) with x₁ = 1.5 to find the root of x³ + x − 5 = 0 to 2 d.p.

x₂ = 1.5183, x₃ = 1.5156, x₄ = 1.5160 → root ≈ 1.52

Exam tip: When squaring both sides of a modulus inequality, both sides must be non-negative (they are, as moduli).

Check yourself

  1. Express (5x + 3)/((x + 1)(x + 3)) in partial fractions.
  2. Solve 2^(x+1) = 7.
  3. State the range of validity of the expansion of (1 + 4x)^(−½).
Show answers
  1. −1/(x + 1) + 6/(x + 3)
  2. x = ln 7/ln 2 − 1 = 1.81
  3. |x| < ¼
Next topicPure 3: further calculus and differential equations

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