Factorise $a^{3} - 3 a^{2} + 3 a + 7$ .

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Solution

Write
$a^{3} - 3 a^{2} + 3 a + 7 = a^{3} - 3 a^{2} + 3 a - 1 + 8 = {(a - 1)}^{3} + 8$
But
${(a - 1)}^{3} + 8 = {(a - 1)}^{3} + 2^{3} = (a - 1 + 2) [{(a - 1)}^{2} - 2 (a - 1) + 4]$
Simplification gives
$[{(a - 1)}^{2} - 2 (a - 1) + 4] = a^{2} - 2 a + 2 - 2 a + 1 + 4 = a^{2} - 4 a + 7$
Thus we obtain
$a^{3} - 3 a^{2} + 3 a + 7 = (a + 1) (a^{2} - 4 a + 7)$ .

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