i am having problem in this proof. i need to find the certain coefficients of this statement on the right side.
given:
$P: \mathbb{C} \Longrightarrow \mathbb{C}, \quad P(x) := 12 − 7x + x^2$
$Q : \mathbb{C} \Longrightarrow \mathbb{C}, \quad Q(x) := 30 + 22x + 10x^2 + 2x^3$
i need to find the coefficients of $c_i$ für $0 \leq i \leq 10$, so that $(P + Q)(x) = \sum_{i=0}^{10}c_i(x-2)^i$.
what i did is: i added $P+Q$ which gave me $2x^3+11x^2+15x+42$ and now i am trying to find a combination of coefficients for $c_i(x-2)^i$ so that if i sum them, it should be equal to $P+Q$. but it seems very tedious, because i need to test for all numbers and sum them. is there any other lighter way for this?
thanks a lot