Which of the following describes the zeroes of the graph of [tex]f(x)=-x^5+9 x^4-18 x^3[/tex]?
A. 0 with multiplicity [tex]3,-3[/tex] with multiplicity 2, and -2 with multiplicity 1
B. 0 with multiplicity 3, 3 with multiplicity 1, and 6 with multiplicity 1
C. 0 with multiplicity 3, 3 with multiplicity 2, and 2 with multiplicity 1
D. 0 with multiplicity [tex]3,-3[/tex] with multiplicity 1, and -6 with multiplicity 1
Answer (2)
The zeroes of the function f ( x ) = − x 5 + 9 x 4 − 18 x 3 are determined by factoring the polynomial, which results in zeroes at 0 (multiplicity 3), 3 (multiplicity 1), and 6 (multiplicity 1). The correct choice, therefore, is option B: 0 with multiplicity 3, 3 with multiplicity 1, and 6 with multiplicity 1.
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Factor the polynomial f ( x ) = − x 5 + 9 x 4 − 18 x 3 to get f ( x ) = − x 3 ( x − 3 ) ( x − 6 ) .
Identify the zeroes by setting each factor to zero: x = 0 , x = 3 , and x = 6 .
Determine the multiplicity of each zero from the exponent of its corresponding factor.
State the zeroes and their multiplicities: 0 with multiplicity 3, 3 with multiplicity 1, and 6 with multiplicity 1. The final answer is 0 with multiplicity 3, 3 with multiplicity 1, and 6 with multiplicity 1, which corresponds to the second option.
0 with multiplicity 3 , 3 with multiplicity 1 , and 6 with multiplicity 1
Explanation
Understanding the Problem We are given the function f ( x ) = − x 5 + 9 x 4 − 18 x 3 and asked to describe its zeroes and their multiplicities. To do this, we need to factor the polynomial.
Factoring the Polynomial First, we factor out the common factor of − x 3 from each term: f ( x ) = − x 3 ( x 2 − 9 x + 18 ) Now, we need to factor the quadratic x 2 − 9 x + 18 . We are looking for two numbers that multiply to 18 and add to -9. These numbers are -3 and -6. So, we can factor the quadratic as ( x − 3 ) ( x − 6 ) .
Thus, the factored form of the function is: f ( x ) = − x 3 ( x − 3 ) ( x − 6 )
Finding the Zeroes To find the zeroes of the function, we set f ( x ) = 0 :
− x 3 ( x − 3 ) ( x − 6 ) = 0 This gives us the zeroes x = 0 , x = 3 , and x = 6 .
Determining Multiplicities Now, we determine the multiplicity of each zero. The multiplicity of a zero is the power of the corresponding factor in the factored form of the polynomial.
The factor − x 3 corresponds to the zero x = 0 with multiplicity 3.
The factor ( x − 3 ) corresponds to the zero x = 3 with multiplicity 1.
The factor ( x − 6 ) corresponds to the zero x = 6 with multiplicity 1. Therefore, the zeroes of the function are 0 with multiplicity 3, 3 with multiplicity 1, and 6 with multiplicity 1.
Final Answer The zeroes of the graph of f ( x ) = − x 5 + 9 x 4 − 18 x 3 are 0 with multiplicity 3, 3 with multiplicity 1, and 6 with multiplicity 1.
Examples
Understanding the zeroes and their multiplicities is crucial in various fields. For instance, in engineering, analyzing the stability of a system often involves finding the roots of a characteristic equation, where the roots represent the system's modes. The multiplicity of a root can indicate the system's sensitivity to certain disturbances. Similarly, in physics, when studying wave phenomena, the zeroes of a wave function and their multiplicities can reveal important information about the wave's behavior and properties, such as nodes and antinodes in standing waves. In computer graphics, polynomial functions are used to model curves and surfaces, and understanding their zeroes helps in rendering and manipulating these shapes effectively.
