<s>[INST] Convert Polynomial to Sum I have to convert the polynomial: $P_4=x^4+7x^3-13x^2-103x-84$ into the form: $$P_4(x)=\sum_{i=0}^4 a_i(x-1)^i$$ using Horner's Method.
I can evaluate both of them using Horner's Method, but I can't figure out how to convert them into one another.
 [/INST] Note that:
$$a_0+a_1(x-1)+a_2(x-1)^2+a_3(x-1)^3+a_4(x-1)^4=x^4+7x^3-13x^2-103x-84 \stackrel{x=1}{\Rightarrow} \\
\color{red}{a_0}=1+7-13-103-84=\color{red}{-192};\\
(x-1)[a_1+a_2(x-1)+a_3(x-1)^2+a_4(x-1)^3]=x^4+7x^3-13x^2-103x+108 \stackrel{\div (x-1)}{\Rightarrow} \\
a_1+a_2(x-1)+a_3(x-1)^2+a_4(x-1)^3=x^3+8x^2-5x-108 \stackrel{x=1}{\Rightarrow} \\
\color{red}{a_1}=1+8-5-108=\color{red}{-104};\\
(x-1)[a_2+a_3(x-1)+a_4(x-1)^2]=x^3+8x^2-5x-4 \stackrel{\div (x-1)}{\Rightarrow} \\
a_2+a_3(x-1)+a_4(x-1)=x^2+9x+4 \stackrel{x=1}{\Rightarrow} \\
\color{red}{a_2}=1+9+4=\color{red}{14};\\
\vdots$$
Can you continue and finish?
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