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.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