Final Answer:
The polynomial function ( p ) is given by ( p(x) = 3x(x-1) ).
Step-by-step explanation:
The polynomial function ( p(x) ) is presented in factored form, where ( 3x ) and ( (x-1) ) are the factors. In this form, it is easy to identify the roots or zeros of the polynomial by setting each factor equal to zero.
Setting ( 3x = 0 ) gives ( x = 0 ) as one root, and setting ( (x-1) = 0 ) gives ( x = 1 ) as the other root. These are the points where the polynomial intersects the x-axis.
The general form of the polynomial, ( p(x) = 3x(x-1) ), tells us that the degree of the polynomial is 2, and it is factored into linear factors. This form provides valuable information about the behavior of the function, such as the roots and the leading coefficient.
Understanding polynomial functions and their factored forms is crucial in various mathematical applications, including algebraic manipulations, graphing, and solving equations.