Which is the complete factorisation of $24x^{4} - 36x^{3} + 12x^{2}$ ?

Which is the complete factorisation of $24x^{4} - 36x^{3} + 12x^{2}$ ?
  1. $12x^{2}(2x^{2} - 3x + 1)$
  2. $12x^{2}(2x - 1)(x - 1)$
  3. $12x^{2}(2x + 1)(x - 1)$
  4. $12x(2x^{3} - 3x^{2} + x)$

Solution

Common factor $12x^{2}$ gives $12x^{2}(2x^{2} - 3x + 1)$. Splitting $-3x = -2x - x$: $2x^{2} - 2x - x + 1 = 2x(x-1) - (x-1) = (2x-1)(x-1)$. So full factorisation is $12x^{2}(2x-1)(x-1)$.

Asked in: IMO

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