Which of the following describes the zeroes of the graph of [tex]f(x)=3 x^6+30 x^5+75 x^4[/tex]?
A. -5 with multiplicity 2 and [tex]\frac{1}{3}[/tex] with multiplicity 4
B. 5 with multiplicity 2 and [tex]\frac{1}{3}[/tex] with multiplicity 4
C. -5 with multiplicity 2 and 0 with multiplicity 4
D. 5 with multiplicity 2 and 0 with multiplicity 4
Answer (1)
Factor the polynomial: f ( x ) = 3 x 4 ( x + 5 ) 2 .
Identify the zeroes by setting each factor to zero: x 4 = 0 and ( x + 5 ) 2 = 0 .
Determine the zeroes: x = 0 and x = − 5 .
State the zeroes and their multiplicities: -5 with multiplicity 2 and 0 with multiplicity 4. -5 with multiplicity 2 and 0 with multiplicity 4
Explanation
Understanding the Problem We are given the function f ( x ) = 3 x 6 + 30 x 5 + 75 x 4 and asked to find its zeroes and their multiplicities. This means we need to find the values of x for which f ( x ) = 0 , and determine how many times each zero appears as a root of the polynomial.
Factoring the Polynomial To find the zeroes, we first factor the polynomial. We can factor out 3 x 4 from each term: f ( x ) = 3 x 4 ( x 2 + 10 x + 25 ) Now, we can further factor the quadratic expression: f ( x ) = 3 x 4 ( x + 5 ) ( x + 5 ) = 3 x 4 ( x + 5 ) 2
Finding the Zeroes Now we set f ( x ) = 0 to find the zeroes: 3 x 4 ( x + 5 ) 2 = 0 This equation is satisfied if x 4 = 0 or ( x + 5 ) 2 = 0 .
Determining Multiplicities From x 4 = 0 , we get x = 0 . Since the exponent is 4, the multiplicity of this zero is 4. From ( x + 5 ) 2 = 0 , we get x + 5 = 0 , so x = − 5 . Since the exponent is 2, the multiplicity of this zero is 2.
Final Answer Therefore, the zeroes of the function are -5 with multiplicity 2 and 0 with multiplicity 4.
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. The roots represent the system's natural frequencies, and their multiplicities can indicate the system's behavior near those frequencies. Similarly, in physics, when studying wave phenomena, the zeroes of a wave function can represent points of destructive interference, and their multiplicities can provide insights into the nature of the interference pattern. By understanding the zeroes and their multiplicities, engineers and physicists can design more stable systems and predict wave behavior more accurately.
