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Question
Using binomial theorem, write down the expansions :
(ii) \[\left( 2x - 3y \right)^4\]
Solution
(ii) (2x − 3y)4
\[= ^{4}{}{C}_0 (2x )^4 (3y )^0 - ^{4}{}{C}_1 (2x )^3 (3y )^1 + ^{4}{}{C}_2 (2x )^2 (3y )^2 - ^{4}{}{C}_3 (2x )^1 (3y )^3 + ^{4}{}{C}_4 (2x )^0 (3y )^4 \]
\[ = 16 x^4 - 4 \times 8 x^3 \times 3y + 6 \times 4 x^2 \times 9 y^2 - 4 \times 2x \times 27 y^3 + 81 y^4 \]
\[ = 16 x^4 - 96 x^3 y + 216 x^2 y^2 - 216x y^3 + 81 y^4\]
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\[= ^{5}{}{C}_0 (2x )^5 (3y )^0 +^{5}{}{C}_1 (2x )^4 (3y )^1 + ^{5}{}{C}_2 (2x )^3 (3y )^2 + ^{5}{}{C}_3 (2x )^2 (3y )^3 + ^{5}{}{C}_4 (2x )^1 (3y )^4 +^{5}{}{C}_5 (2x )^0 (3y )^5\]
\[= 32 x^5 + 5 \times 16 x^4 \times 3y + 10 \times 8 x^3 \times 9 y^2 + 10 \times 4 x^2 \times 27 y^3 + 5 \times 2x \times 81 y^4 + 243 y^5 \]
\[ = 32 x^5 + 240 x^4 y + 720 x^3 y^2 + 1080 x^2 y^3 + 810x y^4 + 243 y^5 \]
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