Question

The term independent of $x$ in the expansion $(1+2x+\frac{2}{x}{)}^3$ is

• A. 12
• B. 18
• C. 36
• D. 25

Answer 1

The correct option is D 25

The above question can be written as
$(1+2(x+\frac{1}{x}\left)\right){}^3$
$=1+8(x+\frac{1}{x}){}^3+6(x+\frac{1}{x})+12(x+\frac{1}{x}){}^2$
$=1+8(x^3+3x+\frac{3}{x}+\frac{1}{x^3})+6(x+\frac{1}{x})+12(x^2+2+\frac{1}{x^2})$
$=8x^3+\frac{8}{x^3}+12x^2+\frac{12}{x^2}+30x+\frac{30}{x}+25$ ...(i)
Hence from (i), the term independent of $x$ is $25$