Question

Find the constant term (term independent of x) in the expansion of ${\displaystyle {(x-\frac{2}{x^2})}^{15}}$

Answer 1

Here, a = x, b = ${\displaystyle \frac{-2}{x^2}}$, n = 15

We have, t_r+1 = ⁿC_r a^n–r .b^r 

= ${\displaystyle {}^{15}{\text{C}}_{\text{r}}{\left(x\right)}^{15-\text{r}}\cdot {\left(\frac{-2}{x^2}\right)}^{\text{r}}}$

= ¹⁵C_r x^15–r.(–2)^r. x^–2r

= ¹⁵C_r (–2)^r x^15–3r

To get the term independent of x, we must have

x^15–3r = x⁰

∴ 15 – 3r = 0

∴ r = 5

∴ the term independent of x

= ¹⁵C₅ (– 2)⁵

= ${\displaystyle \frac{15!}{5!10!}{(-2)}^5}$

= ${\displaystyle \frac{15\times 14\times 13\times 12\times 11}{5\times 4\times 3\times 2\times 1}\times (-32)}$

= – 96096

∴ the term independent of x is – 96096.