Question

. Find the term independent of x in the expansion of (x^ 3+2/x)^ 20

Answer 1

By the Binomial Theorem

$$(a+b)^n=\dbinom{n}{0}a^nb^0+\dbinom{n}{1}a^{n-1}b^{1}+\dbinom{n}{2}a^{n-2}b^{2}$$

$$+...+\dbinom{n}{n-1}a^{1}b^{n-1}+\dbinom{n}{n}a^{0}b^{n}$$

We have $a=x^3, b=\dfrac{2}{x}=2x^{-1}, n=20.$

If the term is independent of $x,$ then

$$(x^3)^{20-k}(x^{-1})^k=x^{0},$$

$$60-3k-k=0$$

$$k=15$$

$$\dbinom{20}{15}(x^3)^{20-15}\bigg(\dfrac{2}{x}\bigg)^{15}=15504(32768)$$

$$=508035072$$