<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] This is similar to changing base of a number system.
So you can use long division.
$$P(x) = x^4 + 2x^3-3x^2-4x+1$$
To express it as powers of $(x+1)$, divide $P(x)$ by the new base $(x+1)$
$$\begin{align}
x^4 + 2x^3-3x^2-4x+1 &= (x+1)(x^3+x^2-4x) + \color{blue}{1} \\
x^3+x^2-4x &= (x+1)(x^2-4)+ \color{blue}{4}\\  
x^2-4 &= (x+1)(x-1) \color{blue}{-3}\\  
x-1 &= (x+1)(1)  \color{blue}{- 2}\\  
1 &= (x+1)0+  \color{blue}{1}\\
\end{align}$$ 
$$P(x) = \color{blue}{1}(x+1)^4 \color{blue}{-2}(x+1)^3\color{blue}{-3}(x+1)^2+\color{blue}{4}(x+1)+\color{blue}{1}$$ 
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