Discussion

The constant term in the binomial expansion of \(\left(2x - \frac{5}{x^{2}}\right)^{6}\) is

Answer: E

The constant term is 6000.

The general term of \(\left(2x - \frac{5}{x^{2}}\right)^{6}\) is

\[ \begin{aligned} T_{r+1} \\ &= {}^{6}C_{r}(2x)^{6-r}\left(-\frac{5}{x^{2}}\right)^{r} \end{aligned} \]

with power of \(x\) equal to \(6 - 3r\). This is zero when \(r = 2\):

\[ T_{3} = 15 \times 2^{4} \times 25 = 6000 \]

The sign is positive because \((-5)^{2}\) is positive. The other options come from using the wrong \(r\) or dropping a factor.

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