Find a power series representation of $\frac{8 - x}{x^2 + x + 1}.$

Question

I have the next function. And I need to find his power series representation over reals $$ f(x)= \frac{8 - x}{x^2 + x + 1}$$ I can't find a way in which I can use this: $$ \frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$$ So any help you could give will be appreciated.

Answer 1

Hint

$$f(x)= \frac{8 - x}{x^2 + x + 1}=\frac{(8 - x)(1-x)}{1-x^3}$$

Answer 2

I must confess that I do not see why you would like to use $$\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$$ Just perform the long division to get $$\frac{8 - x}{1+x+x^2}=8-9 x+x^2+8 x^3-9 x^4+x^5+8 x^6-9 x^7+x^8+8 x^9-9 x^{10}+x^{11}+8 x^{12}-9 x^{13}+x^{14}+8 x^{15}+O\left(x^{16}\right)$$ in which you can notice interesting patterns.