- Identify the required roots as \(-4\), \(2\), and \(i\).
- Apply the Complex Conjugate Root Theorem, which requires \(-i\) to be a root as well.
- Construct the factored form of the polynomial using the roots and the leading coefficient \(3\): \(f(x) = 3(x+4)(x-i)(x+i)(x-2)\).
- Conclude that the correct polynomial is \(\boxed{f(x) = 3(x+4)(x-i)(x+i)(x-2)}\).
Explanation
- Analyzing the Given Roots and Theorem Requirements
The objective is to identify a polynomial function that has a leading coefficient of \(3\) and roots at \(-4\), \(i\), and \(2\), each with multiplicity of \(1\). According to the Fundamental Theorem of Algebra, a polynomial of degree \(n\) has exactly \(n\) roots (counting multiplicities). Given the roots \(-4\), \(i\), and \(2\), we must consider the Complex Conjugate Root Theorem. This theorem states that for a polynomial with real coefficients, if \(a + bi\) is a root, then its conjugate \(a - bi\) must also be a root. Thus, since \(i = 0 + 1i\) is a root, \(-i = 0 - 1i\) must also be a root.
- Constructing the Polynomial Function
With the roots identified as \(-4\), \(i\), \(-i\), and \(2\), we can express the polynomial in its factored form. The factors corresponding to these roots are \((x - (-4)) = (x + 4)\), \((x - i)\), \((x - (-i)) = (x + i)\), and \((x - 2)\). Multiplying these factors gives a polynomial with a leading coefficient of \(1\). Since the problem states the leading coefficient must be \(3\), we multiply the entire expression by \(3\). The polynomial is thus:
\[f(x) = 3(x + 4)(x - i)(x + i)(x - 2)\]
- Identifying the Correct Polynomial Option
Comparing our constructed polynomial \(f(x) = 3(x+4)(x-i)(x+i)(x-2)\) with the provided options:
- \(f(x) = 3(x+4)(x-1)(x-2)\)
- \(f(x) = (x-3)(x+4)(x-1)(x-2)\)
- \(f(x) = (x-3)(x+4)(x-i)(x+i)(x-2)\)
- \(f(x) = 3(x+4)(x-i)(x+i)(x-2)\)
The last option matches our derived polynomial exactly. Therefore, the correct polynomial is \(f(x) = 3(x+4)(x-i)(x+i)(x-2)\).
Examples
Polynomials are essential in fields like engineering and physics for modeling complex systems. For instance, if you are designing a control system for a drone, you might use a polynomial whose roots represent the "poles" of the system. Choosing the right polynomial helps ensure the system is stable. Understanding the relationship between roots and factors allows designers to tailor the behavior of the system to meet specific stability requirements, such as ensuring that the response to external disturbances damps out over time rather than growing indefinitely.