What is the complete factorisation of x³ − 9x?

Correct answer: x(x − 3)(x + 3)

The correct answer is C: x(x − 3)(x + 3). Begin by taking the common factor x from both terms: x³ − 9x = x(x² − 9). The remaining expression is a difference of squares because 9 = 3². Therefore x² − 9 = x² − 3² = (x − 3)(x + 3). Combining the common factor with these two factors gives x(x − 3)(x + 3). Option A is incomplete and incorrect: x(x − 9) expands to x² − 9x, not x³ − 9x. Option B is incorrect because (x − 3)² = x² − 6x + 9, which introduces an unwanted middle term when multiplied by x. Option C is correct; multiplying (x − 3)(x + 3) gives x² − 9, and then multiplying by x gives x³ − 9x. Option D omits the common factor x, so it expands only to x² − 9 and cannot equal the cubic expression. A reliable routine is to remove the common factor first, then look for a familiar identity, and finally expand the answer to verify it. Memory cue: factor out x before recognising x² − 9 as a difference of squares.