To find the zeros of the polynomial function [tex]\( f(x) = x^3 + 25x^2 + 233x + 801 \)[/tex], we'll determine where [tex]\( f(x) = 0 \)[/tex]. Let's solve this step-by-step.
Let's start by considering the given polynomial:
[tex]\[ f(x) = x^3 + 25x^2 + 233x + 801 \][/tex]
We need to find the solutions for when [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ x^3 + 25x^2 + 233x + 801 = 0 \][/tex]
The roots of this polynomial can be found by solving this equation. The solutions to this polynomial are:
[tex]\[ x = -9, \quad x = -8 - 5i, \quad x = -8 + 5i \][/tex]
These solutions consist of one real root and two complex conjugate roots. To summarize the zeros of the function:
- One real root: [tex]\( x = -9 \)[/tex]
- Two complex roots: [tex]\( x = -8 - 5i \)[/tex] and [tex]\( x = -8 + 5i \)[/tex]
So, the polynomial [tex]\( f(x) = x^3 + 25x^2 + 233x + 801 \)[/tex] has the zeros [tex]\( x = -9, -8 - 5i, \)[/tex] and [tex]\( -8 + 5i \)[/tex].
