This problem and solution are in the book. I need help understanding the solution.
Problem: Let u be a root of the polynomial $x^3+3x+3$. In $\mathbb Q(u)$, express $(7-2u+u^2)^{-1}$ in the form $a + bu+cu^2$
Solution: Dividing $x^3+3x+3$ by $x^2-2x+7$ gives the quotient $x+2$ and remainder -11. Thus $u^3+3u+3=(u+2)(u^2-2u+7)-11$ . So $(7-2u+u^2)^{-1}=(2/11)+(1/11)u$
My question: $1=(1/11)(u+2)(u^2-2u+7)-(1/11)(u^3+3u+3)$. And $(1/11)(u+2)(u^2-2u+7)=1$ to get the inverse as above. But WHY?? Clearly 1= (...) - (...) sum of 2 terms, how can 1 = the first term only? What is the logic here? Please, help
The logic is that you are not working with $x$ the variable, you are working with $u$ a complex number that satisfies $u^3+3u+3=0$.