Consider the general non-zero polynomial function $P(x) = a_nx^n + a_{n-1}x^{n-1} +  a_1x +a_0$. ($a_n \ne 0$)

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

The number $a_n$ is referred to as the "leading coefficient" and the value of $P(0) = a_0$ gives the second coordinate of the Y-intercept, $(0,a_0)$ for the graph of $P$.

The Factored Form:

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 :
$P(x) = A_P*P_1(x)*P_2(x)*...*P_m(x)*g_P(x)$ .

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 $A_P$.

Connection between the Standard and Factored Form when $g_f = 1$.
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 $P$.

Example OAF.PFF.1 : $P(x) = 2*(x-2)*(x+1)$.

Example OAF.PFF.2 $P(x) =\frac 1 2*(x-2)^2*(x+1)^3$.

Example OAF.PFF.3 : $P(x) =\frac 1 2*(x^2+1)*(x-2)*(x+1)$.

Example OAF.PFF.4 : $P(x) =\frac 1 2*x^2*(x-2)^2*(x+1)^3$.

Example OAF.DPFF.0 : Dynamic Visualization of Factored Form for a Polynomial Function: Graphs and Mapping Diagrams

Example OAF.DPSF.0 : Dynamic Visualization of standard form for a polynomial function: Graphs and Mapping Diagrams