This form for a polynomial function is usually described as **the
"standard polynomial"
form of $P$. **

There is one other very useful form for the polynomial function $P$ that uses $A= a_n$, the leading coefficient of $P$, and the fundamental theorem of algebra to understand $P(x)$ as a product.

In this form, $P$ is expressed as a product of powers of linear functions, $P_j(x) =(x- r_j)^{p_j}$, with $ j = 1,2, ... , m$, and a monic polynomial $g_P$ which has no real roots :

This form is called the "factored" form of $P$. From the factor theorem of
polynomial algebra, $P(x) = 0$ if and only if $x = r_j$ for
some $j$.** At times for convenience we'll use $A$ for $A_P$.**

Expanding the factored form expression gives $P(x) =A_P*P_1(x)*P_2(x)*...*P_m(x) = a_nx^n + a_{n-1}x^{n-1} + .... + a_1x + a_0$.

From this expression we can see the connection between the coefficients in the standard form and the numbers $r_j$ and {p_j} of the factored form.

In the standard form the coefficient of $x^{n-1}$ is $a_{n-1}$, but from expanding the factored form we have $a_{n-1} = A_P*(p_1*r_1+p_2*r_2 + ... +p_n*r_n)$ and $a_0 =A_P*r_1^{p_1}* r_2^{p_2}* ... * r_n^{p_n}$.

The importance of the roots of $P$ to the general behavior of $P$ near the roots is covered thoroughly in most 2nd and advanced courses in algebra.

In later subsections we explore how mapping diagrams for polynomial functions help us visualize these forms and understand further the meaning of the shape of the graph of $P$.