<s>[INST] Using Horner's Method I'm trying to evaluate a polynomial recursively using Horner's method.
It's rather simple when I have every value of $x$ (like: $x+x^2+x^3...$), but what if I'm missing some of those? Example: $-6+20x-10x^2+2x^4-7x^5+6x^7$.
I would also appreciate it if someone could explain the method in more detail, I've used the description listed here but would like some more explanation.
 [/INST] Horner's method or form is also sometimes called nested form.  You can think of it as starting with the whole polynomial $$6x^7-7x^5+2x^4-10x^2+20x-6,$$ setting aside the constant term (if it's zero, you can set aside a zero here) and factoring out an $x$ from the remaining terms $$(6x^6-7x^4+2x^3-10x+20)x-6,$$ and then repeating the process for the innermost polynomial (the part that isn't yet fully in nested form:
$$((6x^5-7x^3+2x^2-10)x+20)x-6$$
$$(((6x^4-7x^2+2x)x-10)x+20)x-6$$
$$((((6x^3-7x+2)x+0)x-10)x+20)x-6$$
$$(((((6x^2-7)x+2)x+0)x-10)x+20)x-6$$
$$((((((6x)x-7)x+2)x+0)x-10)x+20)x-6$$
$$(((((((6)x+0)x-7)x+2)x+0)x-10)x+20)x-6$$
Alternately, if your polynomial is $$p(x)=\sum_{k=0}^{n}a_kx^k=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$$ you can think of $p(x)$ in nested form as $p_0$ where $p_n=a_n$ and $p_{k-1}=p_kx+a_{k-1}$.  That is, start with the leading coefficient ($a_n$).  Multiply by $x$ and add the next coefficient (adding zero to fill in for "missing" powers of $x$), repeating until you've added the constant term ($a_0$). Back to the same example:
$$\begin{align}
p_7&=6\\
p_6&=(6)x+0\\
p_5&=((6)x+0)x-7\\
p_4&=(((6)x+0)x-7)+2\\
p_3&=((((6)x+0)x-7)+2)x+0\\
p_2&=(((((6)x+0)x-7)+2)x+0)-10\\
p_1&=((((((6)x+0)x-7)+2)x+0)-10)+20\\
p_0&=(((((((6)x+0)x-7)+2)x+0)-10)+20)-6
\end{align}$$
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