Question

The coefficient of $x^7$ in the expansion of ${(\frac{x^2}{2}-\frac{2}{x})}^8$ is:

• A. $56$
• B. $-56$
• C. $14$
• D. $-14$
• E. $0$

Answer 1

The correct option is D $-14$

${(\frac{x^2}{2}-\frac{2}{x})}^8$ $=\frac{1}{(2x{)}^8}(x^3-4{)}^8$
$=\frac{1}{\left(2x^8\right)}\left[\right(x^3{)}^8+8\left(x^3{)}^7\right(-4)+\frac{8.7}{1.2}(x^3{)}^6(-4{)}^2+\frac{\mathrm{8.7.6}}{\mathrm{1.2.3}}(x^3{)}^5(-4{)}^3+.....].$
The required term is $\frac{1}{2^8.x^8}C(8,5)\left(x^3{)}^5\right(-4{)}^3=\frac{1}{\mathrm{8.7.6}}\mathrm{1.2.3}(-4{)}^3{x}^7=-14x^7;$
The required coefficient is $-14$.