Find the expansion of (3x^2 - 2ax + 3a^2)^3 using binomial theore… - Vidyadip

Question

Find the expansion of $(3x^2 - 2ax + 3a^2)^3$ using binomial theorem.

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Answer

We have
$(3x^2 - 2ax + 3a^2)^3 = [(3x^2 - 2ax) + 3a^2)]^3$
${ = ^3}{C_0}{(3{x^2} - 2ax)^3}{ + ^3}{C_1}{(3{x^2} - 2ax)^2}$$(3{a^2}){ + ^3}{C_2}(3{x^2} - 2ax){(3{a^2})^2}{ + ^3}{C_3}{(3{a^2})^3}$
$ = {(3{x^2} - 2ax)^3} + 3 \times 3{a^2}{(3{x^2} - 2ax)^2}$$ + 3 \times 9{a^4}(3{x^2} - 2ax) + 27{a^6}$
$ = (27{x^6} - 8{a^3}{x^3} - 54a{x^5} + 36{a^2}{x^4})$$ + 9{a^2}(9{x^4} + 4{a^2}{x^2} - 12a{x^3})$$ + 27{a^4}(3{x^2} - 2ax) + 27{a^6}$
$ = 27{x^6} - 8{a^3}{x^3} - 54a{x^5} + 36{a^2}{x^4}$$ + 81{a^2}{x^4} + 36{a^4}{x^2} - 108{a^3}{x^3} + 81{a^4}{x^2}$$ - 54{a^5}x + 27{a^6}$
$ = 27{x^6} - 54a{x^5} + 117{a^2}{x^4} - 116{a^3}{x^3}$$ + 117{a^4}{x^2} - 54{a^5}x27{a^6}$

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