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Binomial Theorem

Find the expa...

Question

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

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Solution

$(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$
$(3 x^{2} - 2 a x + 3 a^{2})^{3}$ $= [3 x^{2} - a (2 x - 3 a)]^{3}$
$= (3 x^{2})^{3} - 3 (3 x^{2})^{2} . a (2 x - 30) + 3 \times 3 x^{2} a^{2} (2 x - 3 a)^{2} - a^{3} (2 x - 3 a)^{3}$
$= 27 x^{6} - 27 x^{4} a (2 x - 3 a) + a x^{2} a^{2} (4 x^{2} - 12 a x + 9 a^{2}) - a^{3} (8 x^{3} - 3 \times 4 \times x^{2} \times 3 a + 3 \times 4 \times x^{2} \times 3 a + 3 \times 2 x \times 9 a^{2} - 27 + a^{3})$
$= 27 x^{6} - 54 x^{5} a + 81 a^{2} x^{4} - 108 x^{3} a^{3} + 81 a^{4} x^{2} - 8 a^{3} x^{3} + 36 a^{4} x^{2} - 54 x a^{5} + 27 a^{6}$
= $27 x^{6} - 54 x^{5} a + 117 a^{2} x^{4} - 116 a^{3} x^{3} + 117 a^{4} x^{2} - 54 a^{5} x + 27 a^{6}$

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