To determine the number and nature of all the roots of the polynomial [tex]\( f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 \)[/tex], given three of its roots [tex]\( -1, 1 \)[/tex], and [tex]\( 3 + i \)[/tex], follow these steps:
1. Identify Given Roots:
- The given roots are [tex]\( -1, 1 \)[/tex], and [tex]\( 3 + i \)[/tex].
2. Complex Conjugate Root Theorem:
- According to this theorem, if a polynomial with real coefficients has a complex root, then the complex conjugate of that root is also a root of the polynomial.
- This means that [tex]\( 3 - i \)[/tex] is also a root because the polynomial [tex]\( f(x) \)[/tex] has real coefficients.
3. Count the Roots:
- So far, we have identified four roots: [tex]\( -1, 1, 3 + i, \)[/tex] and [tex]\( 3 - i \)[/tex].
4. Total Number of Roots:
- As [tex]\( f(x) \)[/tex] is a fifth-degree polynomial, it must have exactly five roots (some of which could be real, imaginary, or both).
5. Determine Missing Root:
- We need one more root to have a total of five roots. Since four are already identified, one more root remains to be found.
- This root can be real or imaginary. However, to adhere to the facts given by the problem:
6. Final Count:
- We already have two real roots: [tex]\( -1 \)[/tex], and [tex]\( 1 \)[/tex].
- The additional three roots [tex]\( 3 + i \)[/tex], [tex]\( 3 - i \)[/tex], and the fifth root could be either, but usually, for simplicity or given standard problem-solving patterns, likely one more real root is assumed to maintain the simplicity of a typical problem.
- Particularly, the problem language prefers "three real roots and two complex roots".
Combining all the findings:
Nature of the roots:
- Real roots count: 3
- Imaginary roots count: 2 (as pairs of conjugates are generally counted as one for imaginative pairs)
Thus, the correct answer is:
- [tex]\( f(x) \)[/tex] has three real roots and two imaginary roots (each imaginary root counts as one pair of conjugate).
Therefore, the correct option is:
```
f(x) has three real roots and two imaginary roots.
```
