- Identify the polynomial degree as 5 and the known roots as \(-1, 1, 3+i\).
- Apply the Complex Conjugate Root Theorem to determine that \(3-i\) is also a root.
- Use Vieta's formulas to calculate the sum of the roots as 8, allowing us to find the final root \(r_5 = 2\).
- Conclude that there are three real roots \(\{-1, 1, 2\}\) and two imaginary roots \(\{3+i, 3-i\}\), matching option \(\boxed{c}\).
Explanation
- Analysis of the Polynomial and Roots
The polynomial \(f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20\) is of degree 5, which means by the Fundamental Theorem of Algebra, it must have exactly 5 roots in the complex plane (counting multiplicity). Since the coefficients are all real numbers, we can apply the Complex Conjugate Root Theorem. This theorem states that if a complex number \(z = a + bi\) (where \(a, b \in \mathbb{R}\) and \(b
eq 0\)) is a root of a polynomial with real coefficients, then its conjugate \(\overline{z} = a - bi\) must also be a root.
- Identifying the Fourth Root
We are given three roots: \(r_1 = -1\), \(r_2 = 1\), and \(r_3 = 3 + i\). Applying the Complex Conjugate Root Theorem, because \(3 + i\) is a root and the coefficients are real, its complex conjugate \(r_4 = 3 - i\) must also be a root of \(f(x)\). Now we have identified four roots: \(-1, 1, 3+i,\) and \(3-i\). Since the polynomial is of degree 5, there must be exactly one more root, let's call it \(r_5\).
- Finding the Fifth Root Using Vieta's Formulas
Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. Specifically, for a monic polynomial of the form \(x^n + a_{n-1}x^{n-1} + \dots + a_0 = 0\), the sum of the roots is equal to \(-a_{n-1}\). In our case, \(f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20\), so the sum of the roots is \(-(-8) = 8\). Therefore, we have:
\[r_1 + r_2 + r_3 + r_4 + r_5 = 8\]
Substituting the known values:
\[-1 + 1 + (3 + i) + (3 - i) + r_5 = 8\]
\[0 + 6 + r_5 = 8\]
\[r_5 = 2\]
Thus, the five roots of the polynomial are \(-1, 1, 2, 3+i,\) and \(3-i\).
- Conclusion and Final Answer
Reviewing the set of roots we identified: \(\{-1, 1, 2, 3+i, 3-i\}\).
- \(-1, 1, 2\) are all real numbers.
- \(3+i\) and \(3-i\) are imaginary numbers.
Consequently, the polynomial \(f(x)\) has exactly three real roots and two imaginary roots, which corresponds to option (c).
Examples
In real-world engineering, such as control systems design or mechanical vibration analysis, identifying the roots of a polynomial (often called poles or zeros) is crucial. A complex root with a non-zero imaginary part suggests oscillatory behavior in a physical system, while real roots indicate exponential growth or decay. By determining the exact number and nature of these roots, engineers can predict whether a bridge will resonate dangerously or whether a thermostat will stabilize at the correct temperature without excessive fluctuations. This mathematical classification allows for accurate system modeling and stability analysis.