Which of the following describes the roots of the polynomial function [tex]$f(x)=(x-3)^4(x+6)^2$[/tex]?
A. -3 with multiplicity 2 and 6 with multiplicity 4
B. -3 with multiplicity 4 and 6 with multiplicity 2
C. 3 with multiplicity 2 and -6 with multiplicity 4
D. 3 with multiplicity 4 and -6 with multiplicity 2
Answer (1)
The polynomial function is f ( x ) = ( x − 3 ) 4 ( x + 6 ) 2 .
Set each factor to zero to find the roots: ( x − 3 ) 4 = 0 and ( x + 6 ) 2 = 0 .
The roots are x = 3 with multiplicity 4 and x = − 6 with multiplicity 2.
The roots of the polynomial are 3 with multiplicity 4 and -6 with multiplicity 2, so the answer is 3 with multiplicity 4 and − 6 with multiplicity 2 .
Explanation
Understanding the Problem We are given the polynomial function f ( x ) = ( x − 3 ) 4 ( x + 6 ) 2 and asked to determine the roots and their multiplicities. The roots of a polynomial are the values of x for which f ( x ) = 0 . The multiplicity of a root is the number of times the corresponding factor appears in the factored form of the polynomial.
Finding the First Root and Its Multiplicity To find the roots, we set each factor of the polynomial to zero. The first factor is ( x − 3 ) 4 . Setting this to zero gives us ( x − 3 ) 4 = 0 , which implies x − 3 = 0 , so x = 3 . Since the factor is raised to the power of 4, the root x = 3 has a multiplicity of 4.
Finding the Second Root and Its Multiplicity The second factor is ( x + 6 ) 2 . Setting this to zero gives us ( x + 6 ) 2 = 0 , which implies x + 6 = 0 , so x = − 6 . Since the factor is raised to the power of 2, the root x = − 6 has a multiplicity of 2.
Stating the Roots and Their Multiplicities Therefore, the roots of the polynomial function f ( x ) = ( x − 3 ) 4 ( x + 6 ) 2 are x = 3 with multiplicity 4 and x = − 6 with multiplicity 2.
Final Answer The roots of the polynomial function f ( x ) = ( x − 3 ) 4 ( x + 6 ) 2 are 3 with multiplicity 4 and -6 with multiplicity 2.
Examples
Understanding the roots and their multiplicities is crucial in various fields. For instance, in physics, when analyzing the behavior of a damped harmonic oscillator, the roots of the characteristic equation determine the system's stability and oscillation patterns. Similarly, in engineering, analyzing the roots of a transfer function helps determine the stability and response of a control system. In computer graphics, understanding polynomial roots is essential for curve and surface modeling, ensuring smooth and predictable shapes.
