<s>[INST] Writing polynomial using powers of (x-a) I have a question related to writing a polynomial using powers of binomial of form $(x-a).$ 
I found an example: polynomial $P(x) = x^4 + 2x^3-3x^2-4x+1$ can be written as 
$ (x+1)^4-2(x+1)^3-3(x+1)^2+4(x+1)+1$ using powers of $(x+1)$ and Horner's Method. How do we obtain this representation of polynomial? How is Horner's Method used for that?
 [/INST] You want an expression $a_4(x+1)^4+a_3(x+1)^3+a_2(x+1)^2+a_1(x+1)+a_0$. Repeatedly apply the method for $x=-1$. The last number in each row will give you the next coefficient starting from the lowest power.
\begin{array}{|c|c|c|c|c|c|}
\hline
& 1 & 2 & -3 & -4 & 1 \\ \hline
-1 & 1& 1& -4 & 0 & 1 \rightarrow a_0\\ \hline
-1 & 1 & 0& -4 & 4 \rightarrow  a_1\\ \hline
-1 & 1 & -1& -3 \rightarrow a_2\\ \hline
\end{array}
and so on.
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