To determine the coefficients [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] of the polynomial function [tex]\( f(x) = x^3 + A x^2 + B x + C \)[/tex], we need additional information about the polynomial such as its roots or specific points that the polynomial passes through. Since this information isn't explicitly provided in the problem statement, we can't determine [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] precisely.
However, I can guide you through the general process of determining the polynomial’s coefficients if the roots or specific points were known.
### Given Roots
If the polynomial has given roots, say [tex]\(\alpha\)[/tex], [tex]\(\beta\)[/tex], and [tex]\(\gamma\)[/tex], then the polynomial can be constructed as:
[tex]\[ f(x) = (x - \alpha)(x - \beta)(x - \gamma) \][/tex]
Expanding this product would give you the coefficients [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex].
### Specific Points
If specific points on the polynomial are given, say [tex]\((x_1, y_1)\)[/tex], [tex]\((x_2, y_2)\)[/tex], and [tex]\((x_3, y_3)\)[/tex], then you can set up a system of equations by substituting these points into the polynomial:
[tex]\[ y_1 = x_1^3 + A x_1^2 + B x_1 + C \][/tex]
[tex]\[ y_2 = x_2^3 + A x_2^2 + B x_2 + C \][/tex]
[tex]\[ y_3 = x_3^3 + A x_3^2 + B x_3 + C \][/tex]
Solving these simultaneous equations will give you the values of [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex].
### General Form
Without specific details on roots or points, the most we can say is the polynomial retains the general cubic form:
[tex]\[ f(x) = x^3 + A x^2 + B x + C \][/tex]
Where:
- [tex]\( A \)[/tex] is the coefficient of [tex]\( x^2 \)[/tex]
- [tex]\( B \)[/tex] is the coefficient of [tex]\( x \)[/tex]
- [tex]\( C \)[/tex] is the constant term
To move forward with actual numbers for [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex], more information about the polynomial would be required.
