Binomial Theorem

Write down the binomial expansion of $\displaystyle \left(\frac{2}{y^2}-5y\right)^3$ and simplify your answer.

(a)Write down and simplify the general term in the binomial expansion of

$\displaystyle \left(2x^2 - \frac{d}{x^3}\right)^7$,

where $\displaystyle d$ is a constant.

(b)Given that the coefficient of $\displaystyle \frac{1}{x}$ is $\displaystyle -70\ 000$, find the value of $\displaystyle d.$

(a)Write down and simplify the general term in the binomial expansion of $\displaystyle \left(3x-\frac{2}{x^2}\right)^8.$

(b)Hence, or otherwise, determine the coefficient of $\displaystyle x^{-1}.$

Use the binomial theorem to expand $\displaystyle \left(\frac{1}{x} - 3x\right)^4.$
Simplify your answer.

Write down the binomial expansion of

$\displaystyle \left(x^{2}-\frac{5}{x}\right)^{4}$

and simplify your answer.