To find all zeros of the given polynomial function [tex]\( f(x) = x^3 + 22x^2 + 177x + 520 \)[/tex], we'll follow these mathematical steps:
1. Understanding Zeros of the Polynomial:
Zeros of a polynomial function are the values of [tex]\( x \)[/tex] for which the function evaluates to zero. In other words, we need to find the roots of the equation:
[tex]\[
x^3 + 22x^2 + 177x + 520 = 0
\][/tex]
2. Apply Polynomial Root-Finding Techniques:
There are several methods to find the roots of a polynomial, such as factoring (if possible), the Rational Root Theorem, synthetic division, and using advanced algebraic methods or numerical methods. For a cubic polynomial, it's generally a mix of analytical and numerical methods.
3. Roots of [tex]\( f(x) \)[/tex]:
Given the polynomial [tex]\( f(x) = x^3 + 22x^2 + 177x + 520 \)[/tex], let's consider the numerical results that show the roots of this cubic equation.
4. Identified Roots:
Based on the given, the roots (zeros) of the polynomial are:
- [tex]\( x = -8 \)[/tex]
- [tex]\( x = -7 - 4i \)[/tex]
- [tex]\( x = -7 + 4i \)[/tex]
5. Multiplicity of the Roots:
For a cubic polynomial like this, each root generally has a multiplicity of 1, since there is no indication of repeated roots from the provided roots. A root with multiplicity [tex]\( k \)[/tex] means that the factor corresponding to the root will be repeated [tex]\( k \)[/tex] times in the factorization of the polynomial.
Thus, the complete list of zeros for the polynomial function [tex]\( f(x) = x^3 + 22x^2 + 177x + 520 \)[/tex] with their appropriate multiplicities is:
- [tex]\( -8 \)[/tex] with multiplicity 1
- [tex]\( -7 - 4i \)[/tex] with multiplicity 1
- [tex]\( -7 + 4i \)[/tex] with multiplicity 1
