Find the zeros of [tex]p(x)=5 x^6(x+2)^7(x-3)^7[/tex]. Also state the multiplicity of each zero of [tex]p(x)[/tex].
Answer (2)
Find the zeros by setting each factor of the polynomial to zero.
Determine the multiplicity of each zero by examining the exponent of its corresponding factor.
The zeros are 0 (multiplicity 6), -2 (multiplicity 7), and 3 (multiplicity 7).
The zeros of p ( x ) are 0 (multiplicity 6), -2 (multiplicity 7), and 3 (multiplicity 7), so the answer is D .
Explanation
Understanding the Problem We are given the polynomial p ( x ) = 5 x 6 ( x + 2 ) 7 ( x − 3 ) 7 . Our goal is to find the zeros of this polynomial and their corresponding multiplicities. The zeros of a polynomial are the values of x for which p ( x ) = 0 . The multiplicity of a zero is the power to which the corresponding factor is raised.
Finding the Zeros To find the zeros, we set p ( x ) = 0 :
5 x 6 ( x + 2 ) 7 ( x − 3 ) 7 = 0
This equation is satisfied if any of the factors are equal to zero.
Determining Multiplicities
The factor 5 x 6 is zero when x = 0 . The exponent is 6, so the multiplicity of the zero x = 0 is 6.
The factor ( x + 2 ) 7 is zero when x + 2 = 0 , which means x = − 2 . The exponent is 7, so the multiplicity of the zero x = − 2 is 7.
The factor ( x − 3 ) 7 is zero when x − 3 = 0 , which means x = 3 . The exponent is 7, so the multiplicity of the zero x = 3 is 7.
Final Answer Therefore, the zeros of p ( x ) are 0 (with multiplicity 6), -2 (with multiplicity 7), and 3 (with multiplicity 7).
Examples
Understanding polynomial zeros and their multiplicities is crucial in various fields, such as engineering and physics. For example, when analyzing the stability of a system, the zeros of a characteristic polynomial determine the system's behavior. A zero with a high multiplicity can indicate a critical point where the system's behavior changes drastically. In circuit analysis, the roots of the impedance function determine the resonant frequencies of the circuit. Similarly, in quantum mechanics, the energy levels of a particle in a potential well are determined by the zeros of a wavefunction. Therefore, mastering the concept of polynomial zeros and their multiplicities is essential for solving real-world problems in science and engineering.
The zeros of the polynomial p ( x ) = 5 x 6 ( x + 2 ) 7 ( x − 3 ) 7 are 0 (multiplicity 6), -2 (multiplicity 7), and 3 (multiplicity 7).
;
