Find the zeros of the polynomial function, and state the multiplicity of each.
[tex]$f(x)=(x+2)^4(x-3)$[/tex]
Answer (1)
Set f ( x ) = 0 to find the zeros of the polynomial.
Solve ( x + 2 ) 4 = 0 to find the zero x = − 2 with multiplicity 4.
Solve ( x − 3 ) = 0 to find the zero x = 3 with multiplicity 1.
The zeros are x = − 2 (multiplicity 4) and x = 3 (multiplicity 1).
Explanation
Understanding the Problem We are given the polynomial function f ( x ) = ( x + 2 ) 4 ( x − 3 ) and asked to find its zeros and their multiplicities.
Setting up the Equation To find the zeros of the polynomial, we need to solve the equation f ( x ) = 0 . This means we need to find the values of x for which ( x + 2 ) 4 ( x − 3 ) = 0 .
Finding the Zeros The equation ( x + 2 ) 4 ( x − 3 ) = 0 is satisfied if either ( x + 2 ) 4 = 0 or ( x − 3 ) = 0 .
Determining Multiplicity of x = -2 If ( x + 2 ) 4 = 0 , then x + 2 = 0 , which gives x = − 2 . The factor ( x + 2 ) appears with an exponent of 4, so the multiplicity of the zero x = − 2 is 4.
Determining Multiplicity of x = 3 If ( x − 3 ) = 0 , then x = 3 . The factor ( x − 3 ) appears with an exponent of 1, so the multiplicity of the zero x = 3 is 1.
Final Answer Therefore, the zeros of the polynomial function are x = − 2 with multiplicity 4 and x = 3 with multiplicity 1.
Examples
Understanding polynomial zeros and their multiplicities is crucial in various fields, such as physics and engineering. For instance, when analyzing the stability of a system, the zeros of a characteristic polynomial determine the system's behavior. A zero with high multiplicity might indicate a critical point where the system's behavior changes drastically. In circuit analysis, the roots of the circuit's transfer function determine the system's stability and response to different input signals. Therefore, understanding the zeros and their multiplicities helps engineers design stable and reliable systems.
