To determine the number and nature of all the roots of the polynomial function [tex]\( f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 \)[/tex], given that three of its roots are [tex]\( -1 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 3+i \)[/tex], let's go step-by-step:
1. Identify the Given Roots:
The problem states that the polynomial [tex]\( f(x) \)[/tex] has three known roots:
[tex]\[
-1, \quad 1, \quad \text{and} \quad 3+i
\][/tex]
2. Consider Complex Conjugate Roots:
Since the coefficients of the polynomial are real, if a polynomial has a complex root, its complex conjugate must also be a root. Therefore, the conjugate of [tex]\( 3+i \)[/tex] is [tex]\( 3-i \)[/tex], which must also be a root.
3. Account for the Complex Conjugate Root:
Now we have the roots:
[tex]\[
-1, \quad 1, \quad 3+i, \quad \text{and} \quad 3-i
\][/tex]
These make up four roots so far.
4. Determine the Fifth Root:
The polynomial [tex]\( f(x) \)[/tex] is of degree 5, which means it must have five roots in total (counting multiplicities). Given the four roots above, there must be one more root.
5. Total Roots:
- Real roots: [tex]\(-1, 1, \text{fifth root}\)[/tex] (we have three real roots in total when we include this unknown fifth root).
- Complex roots: [tex]\(3+i\)[/tex] and [tex]\(3-i\)[/tex] (these are the two imaginary roots).
6. Conclusion:
The polynomial [tex]\( f(x) \)[/tex] has a total of five roots:
- Three real roots.
- Two imaginary roots (complex conjugates).
Based on this analysis, the correct description of the number and nature of all the roots of the polynomial function is:
[tex]\[ f(x) \text{ has three real roots and two imaginary roots.} \][/tex]
