Question

Expand $\left(i\right){\left(1-x+x^2\right)}^4\left(ii\right){\left(1+x+x^2\right)}^3$ in powers of $x$.

Answer 1

This is a power of a trinomial, not a binomial so the binomial theorem does not help much.

However, $1,x,x^2$ are in geometric progression, so we can use a variant of $Pasca{l}^{\prime }striangle$ to find the coefficients we want. Each term in this variant is the sum of the three terms above it. rather than two...

$\left(i\right)$ $(1-x+x^2{)}^4$

$1$

$1$ $1$ $1$

$1$ $2$ $3$ $2$ $1$

$1$ $3$ $6$ $7$ $6$ $3$ $1$

$1$ $4$ $10$ $16$ $19$ $16$ $10$ $4$ $1$

Hence we find : $(1+x+x^2{)}^4=1+4x+10x^2+16x^3+19x^4+16x^5+10x^6+4x^7+x^8$

$\left(ii\right)$ $(1-x+x^2{)}^3$

$1$

$1$ $1$ $1$

$1$ $2$ $3$ $2$ $1$

$1$ $3$ $6$ $7$ $6$ $3$ $1$

Hence we find :

$(1+x+x^2{)}^3=1+3x+6x^2+7x^3+6x^4+3x^5+x^6$