<s>[INST] Polynomial representation Why is the polynomial $P(x)$ represented as
$$ P(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots+ a_2 x^2 + a_1 x + a_0 \text{ ?}$$
A polynomial can be $5x^4 + 3x^3 + 7x^2 + 10x -2$ and it is not necessarily  $a_n x^n + a_{n-1} x^{n-1}+\cdots$ I.e if $ a_n x^n$ is $ 5x^4$, doesn't $a_{n-1} x^{n-1} $ mean $4x^3$?
 [/INST] $$a_{n-1}$$ is not the same as $$a_n -1 $$
The $-1$ is in the subscript in $a_{n-1}$.
Think of it as a function $a$, and we are looking at $a(1), a(2), \dots, a(n-1), a(n)$, so the polynomial is
$$a(n) x^n + a(n-1) x^{n-1} + \dots + a(1) x + a(0)$$
So for instance, if we take $a(m) = m^2$ (can also be written as $a_{m} = m^2$), then for $n=3$ the polynomial will be
$$ 9x^3 + 4x^2 + x $$
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