Find the expansion of ${(3 x^{2} - 2 a x + 3 a^{2})}^{3}$ .

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Solution

${(3 x^{2} - 2 a x + 3 a^{2})}^{3}$

$= {(3 a^{2})}^{3} {1 + \frac{3 x^{2} - 2 a x}{3 a^{2}}}^{3}$
$= {27 a}^{6} {^{3} C_{0} +^{3} C_{1} (\frac{3 x^{2} - 2 a x}{3 a^{2}}) +^{3} C_{2} {(\frac{3 x^{2} - 2 a x}{3 a^{2}})}^{2} +^{3} C_{3} {(\frac{3 x^{2} - 2 a x}{3 a^{2}})}^{3}}$ ...... Using binomial expansion
$= {27 a}^{6} - {54 a}^{5} x + 117 a^{4} x^{2} - 116 a^{3} x^{3} + 117 a^{2} x^{4} - 54 {a x}^{5} + {27 x}^{6}$

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