Let $\mathrm{P}(x)=(1+x)^8(1-x)^6.$ Rewrite $\mathrm{P}(x)$ in the form \( (1-x^2)^6(1+x)^2

Question

Let $\mathrm{P}(x)=(1+x)^8(1-x)^6.$ Rewrite $\mathrm{P}(x)$ in the form $ (1-x^2)^6(1+x)^2. $ Hence find the coefficient of $x^7$. Find the coefficient of $x^8$. Find the sum of all coefficients of $\mathrm{P}(x)$, without fully expanding it.

Accepted Answer

(a) Pair six factors; (b) $-40$; (c) $-5$; (d) $0$.

Binomial Theorem

Binomial Theorem