Pure A: Algebra & Polynomials

Topic overview

Factor theorem, complex factorisation, inequality proofs

Key concepts

Verify STEP 2/3 separately. Complex numbers and 2×2 matrices may belong to STEP 2; De Moivre, general complex roots, planes and vector products, etc. belong to STEP 3.

Key points

◆Follow the prerequisite knowledge requirements given by the examination body; new concepts may only be used when the question provides sufficient guidance.
◆The examples on this site are training with a limited scope and do not represent full syllabus coverage or official difficulty.

Worked examples

Try each example yourself before checking the explanation.

1Written examples within the scope

Let $f(x) = x^3 + px + q$ where $p$ and $q$ are real constants.

Show that $f(x)$ has a repeated root if and only if $4p^3 + 27q^2 = 0$ .

Show that $f(x) = x^3 - 3x - 2$ has a repeated root, and find all roots of $f$ .

Find all values of $k$ for which $x^3 - 3x - 2 = k$ has exactly two distinct real solutions.

Sketch $y = x^3 - 3x - 2$ and use your sketch to determine the number of real solutions of $x^3 - 3x - 2 = k$ for general $k$ .

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