Review question

# Can we find the coefficients of $x^{−12}$ and $x^2$ in this expansion? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9207

## Suggestion

Show that the coefficient of $x^{−12}$ in the expansion of $\left(x^4-\frac{1}{x^2}\right)^5\left(x-\frac{1}{x}\right)^6$ is $−15$, and calculate the coefficient of $x^2$.

Could we expand each of the brackets?

What contribution do we need from each of the bracketed terms to get $x^{-12}$ or $x^2$ terms?

Hence, or otherwise, calculate the coefficients of $x^4$ and $x^{38}$ in the expansion of $(x^2 − 1)^{11}(x^4 + x^2 + 1)^5.$

How are the two parts related? Could we rewrite this new expression to make it look like the first part?

What happens if we mulitply $(x^4+x^2+1)$ by $(x^2-1)$?