One root of a third degree polynomial function [tex]$f(x)$[/tex] is [tex]$-5+2 i$[/tex]. Which statement describes the number and nature of all roots for this function?
A. [tex]$f(x)$[/tex] has two real roots and one imaginary root.
B. [tex]$f(x)$[/tex] has two imaginary roots and one real root.
C. [tex]$f(x)$[/tex] has three imaginary roots.
D. [tex]$f(x)$[/tex] has three real roots.
Answer (2)
The polynomial f ( x ) has one complex root − 5 + 2 i , and therefore must also include its conjugate, − 5 − 2 i , due to the complex conjugate root theorem. This means there are two imaginary roots and one remaining real root. Thus, the answer is B: f ( x ) has two imaginary roots and one real root.
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A third-degree polynomial f ( x ) has one root as − 5 + 2 i .
Complex roots occur in conjugate pairs, so − 5 − 2 i is also a root.
A third-degree polynomial has three roots, so the remaining root must be real.
Therefore, f ( x ) has two imaginary roots and one real root. f ( x ) has two imaginary roots and one real root.
Explanation
Understanding the Problem We are given that f ( x ) is a third-degree polynomial function, and one of its roots is − 5 + 2 i . We need to determine the nature and number of all roots of f ( x ) .
Complex Conjugate Root Theorem Since f ( x ) is a polynomial with real coefficients (this is a standard assumption for such problems, otherwise complex roots wouldn't necessarily come in conjugate pairs), complex roots occur in conjugate pairs. This means that if − 5 + 2 i is a root, then its complex conjugate, − 5 − 2 i , must also be a root.
Determining the Third Root A third-degree polynomial has exactly three roots (counting multiplicity). We already know two roots: − 5 + 2 i and − 5 − 2 i . Since these are complex roots, the remaining root must be a real number. If the third root were also complex, it would need to have a conjugate, resulting in a polynomial of at least degree 4.
Conclusion Therefore, f ( x ) has two imaginary roots ( − 5 + 2 i and − 5 − 2 i ) and one real root.
Examples
Understanding the nature of polynomial roots is crucial in various fields, such as physics and engineering. For instance, when analyzing the stability of a system modeled by a polynomial equation, the roots of the polynomial determine the system's behavior. Complex roots often indicate oscillatory behavior, while real roots indicate stable or unstable states. Knowing the number and nature of roots helps engineers design systems that are stable and perform as expected. For example, in electrical engineering, the roots of the characteristic equation of a circuit determine its stability and response to different inputs. If the roots have positive real parts, the circuit is unstable, leading to oscillations or even damage. Therefore, understanding polynomial roots is essential for designing stable and reliable circuits.
