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Question
Using binomial theorem, write down the expansions :
(x) \[\left( 1 - 2x + 3 x^2 \right)^3\]
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Solution
(x) \[(1 - 2x + 3 x^2 )^3 \]
\[ = ^{3}{}{C}_0 (1 - 2x )^3 + ^{3}{}{C}_1 (1 - 2x )^2 (3 x^2 ) + ^{3}{}{C}_2 (1 - 2x)(3 x^2 )^2 + ^{3}{}{C}_3 (3 x^2 )^3 \]
\[ = (1 - 2x )^3 + 9 x^2 (1 - 2x )^2 + 27 x^4 (1 - 2x) + 27 x^6 \]
\[ = 1 - 8 x^3 + 12 x^2 - 6x + 9 x^2 (1 + 4 x^2 - 4x) + 27 x^4 - 54 x^5 + 27 x^6 \]
\[ = 1 - 8 x^3 + 12 x^2 - 6x + 9 x^2 + 36 x^4 - 36 x^3 + 27 x^4 - 54 x^5 + 27 x^6 \]
\[ = 1 - 6x + 21 x^2 - 44 x^3 + 63 x^4 - 54 x^5 + 27 x^6 \]
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