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

Question

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

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Answer

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

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