To find the zeros of the polynomial
[tex]\[ f(x) = x^5 - 7x^4 + 20x^3 - 68x^2 + 99x - 45, \][/tex]
we will look for values of [tex]\( x \)[/tex] that satisfy [tex]\( f(x) = 0 \)[/tex]. According to the Fundamental Theorem of Algebra, a polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] complex roots (including multiplicity).
### Step-by-Step Solution:
1. Understanding the Polynomial:
- The given polynomial is a fifth-degree polynomial, which means we are looking for a total of five roots.
2. Identifying the Roots:
- Upon finding the roots, we'll observe that they are:
- [tex]\(1\)[/tex]
- [tex]\(5\)[/tex]
- [tex]\( -3i \)[/tex] (where [tex]\(i\)[/tex] is the imaginary unit, with [tex]\(i^2 = -1\)[/tex])
- [tex]\( 3i \)[/tex]
3. Analysis of the Roots:
- The root [tex]\( x = 1 \)[/tex] is a real number.
- The root [tex]\( x = 5 \)[/tex] is a real number.
- The root [tex]\( x = -3i \)[/tex] is an imaginary number.
- The root [tex]\( x = 3i \)[/tex] is an imaginary number.
4. Multiplicity:
- None of these roots appear more than once in our solution, so each has a multiplicity of 1.
So the zeros of the polynomial [tex]\( f(x) = x^5 - 7x^4 + 20x^3 - 68x^2 + 99x - 45 \)[/tex] are:
[tex]\[ \{1, 5, -3i, 3i\} \][/tex]
These zeros include two real roots (1 and 5) and two complex roots (-3i and 3i).
