Question

The expansion of $\left(3 x^2-2 a x+3 a^2\right)^3$ by using binomial theorem is

• A. $37 x^6-74 a x^5+117 a^2 x^4-116 a^3 x^3+117 a^4 x^2-54 a^5 x+27 a^6$
• B. $27 x^6-54 a x^5+117 a^2 x^4-116 a^3 x^3+117 a^4 x^2-54 a^5 x+27 a^6$
• C. $37 x^6+54 a x^5-117 a^2 x^4+116 a^3 x^3+117 a^2 x^4-54 a^2 x^5+27 a^6$
• D. None of the above

Answer 1

Answer: (B)

$\left(3 x^2-2 a x+3 a^2\right)^3=\left[3\left(x^2+a^2\right)-2 a x\right]^3 $
$ ={ }^3 C_0\left\{3\left(x^2+a^2\right)\right\}^3-{ }^3 C_1\left\{3\left(x^2+a^2\right)\right\}^2 2 a x $
$ +{ }^3 C_2 3\left(x^2+a^2\right)(2 a x)^2-{ }^3 C_3(2 a x)^3 $
$=27\left(x^2+a^2\right)^3-3 \times 9\left(x^2+a^2\right)^2 \times 2 a x $
$+3 \times 3\left(x^2+a^2\right) 4 a^2 x^2-8 a^3 x^3$
Now, open the expansion of $\left(x^2+a^2\right)^3,\left(x^2+a^2\right)^2$, we get
$=27\left[\left[^3 C_0\left(x^2\right)^3+{ }^3 C_1\left(x^2\right)^2 a^2+{ }^3 C_2 x^2\left(a^2\right)^2+{ }^3 C_3\left(a^2\right)^3\right]\right. $
$ -27\left[{ }^2 C_0\left(x^2\right)^2+{ }^2 C_1 x^2 a^2+{ }^2 C_2\left(a^2\right)^2\right] \times 2 a x $
$ +9\left(x^2+a^2\right) 4 a^2 x^2-8 a^3 x^3$
$ =27\left[x^6+3 x^4 a^2+3 x^2 a^4+a^6\right]-27\left[x^4+2 x^2 a^2+a^4\right] 2 a x$
$ +36 a^2 x^2\left(x^2+a^2\right)-8 a^3 x^3$
$ =27 x^6+81 x^4 a^2+81 x^2 a^4+27 a^6-54 a x\left(x^4+2 x^2 a^2+a^4\right) $
$ +36 a^2 x^4+36 a^4 x^2-8 a^3 x^3$
$=27 x^6+117 x^4 a^2+117 a^4 x^2+27 a^6$
$-54 a x^5-54 a^5 x-8 a^3 x^3-108 a^3 x^3$
$=27 x^6-54 a x^5+117 a^2 x^4-116 a^3 x^3$
$+117 a^4 x^2-54 a^5 x+27 a^6$