- Use the Factor Theorem to identify factors \((x+4)\), \((x-2)\), \((x-9)\), and \((x+5)^3\) from the given roots and multiplicities.
- Combine these factors into the expression \(f(x) = (x+4)(x-2)(x-9)(x+5)^3\) with a leading coefficient of \(1\).
- Recognize that \((x+5)^3\) is equivalent to \((x+5)(x+5)(x+5)\).
- Identify the correct option matching the constructed expression: \(\boxed{f(x) = (x+5)(x+5)(x+5)(x+4)(x-2)(x-9)}\).
Explanation
- Analyzing the Roots and Multiplicities
To determine the correct polynomial function \(f(x)\), we rely on the Factor Theorem, which states that if \(c\) is a root of a polynomial \(f(x)\), then \((x - c)\) is a factor of the polynomial. When a root has a specific multiplicity \(m\), the corresponding factor \((x - c)\) is raised to the power of \(m\). We are given that the leading coefficient of the polynomial is \(1\) and the roots are:
- \(x = -4\) with multiplicity \(1\), giving the factor \((x - (-4))^1 = (x+4)\).
- \(x = 2\) with multiplicity \(1\), giving the factor \((x - 2)^1 = (x-2)\).
- \(x = 9\) with multiplicity \(1\), giving the factor \((x - 9)^1 = (x-9)\).
- \(x = -5\) with multiplicity \(3\), giving the factor \((x - (-5))^3 = (x+5)^3\).
- Constructing the Polynomial Function
Now, we multiply these factors together to form the polynomial. The general form of the polynomial with these roots is:
\[f(x) = a \cdot (x+4) \cdot (x-2) \cdot (x-9) \cdot (x+5)^3\]
Given that the leading coefficient \(a\) is \(1\), the polynomial becomes:
\[f(x) = (x+4)(x-2)(x-9)(x+5)^3\]
Expanding the term \((x+5)^3\) as \((x+5)(x+5)(x+5)\), we can rewrite the expression as:
\[f(x) = (x+5)(x+5)(x+5)(x+4)(x-2)(x-9)\]
- Final Selection and Conclusion
Comparing our constructed polynomial \(f(x) = (x+5)(x+5)(x+5)(x+4)(x-2)(x-9)\) with the given options, it perfectly matches the third option provided:
- Option 1: \(3(x+5)(x+4)(x-2)(x-9)\) (Incorrect leading coefficient and multiplicity of -5)
- Option 2: \(3(x-5)(x-4)(x+2)(x+9)\) (Incorrect roots and coefficient)
- Option 3: \((x+5)(x+5)(x+5)(x+4)(x-2)(x-9)\) (Correct)
- Option 4: \((x-5)(x-5)(x-5)(x-4)(x+2)(x+9)\) (Incorrect roots)
Thus, the correct polynomial function is \(\boxed{f(x) = (x+5)(x+5)(x+5)(x+4)(x-2)(x-9)}\).
Examples
In engineering, identifying the correct polynomial model for a system's behavior is crucial. For instance, when analyzing the stability of a control system, we often represent the system using its transfer function, where the denominator is a polynomial. Knowing the roots (poles) and their multiplicities allows engineers to determine if the system will oscillate, decay, or become unstable. By constructing these polynomials accurately, designers can ensure that complex machines like drones or robotic arms perform reliably.