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 (2)
The zeroes of the function f ( x ) = 3 x 6 + 30 x 5 + 75 x 4 can be found by factoring it, resulting in − 5 with multiplicity 2 and 0 with multiplicity 4. The correct answer is option C. This means the graph of the function touches the x-axis at − 5 and crosses it at 0 .
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Factor the polynomial: f ( x ) = 3 x 4 ( x + 5 ) 2 .
Find the zeroes by setting each factor to zero: x 4 = 0 and ( x + 5 ) 2 = 0 .
Determine the zeroes and their multiplicities: x = 0 with multiplicity 4 and x = − 5 with multiplicity 2.
State the zeroes and their multiplicities: − 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 the zeroes of the graph of f ( x ) and their multiplicities. The zeroes of the graph are the values of x for which f ( x ) = 0 . To find these, we need to factor the polynomial.
Factoring the Polynomial First, we factor the polynomial f ( x ) = 3 x 6 + 30 x 5 + 75 x 4 . We can factor out 3 x 4 from each term: f ( x ) = 3 x 4 ( x 2 + 10 x + 25 ) Now, we factor the quadratic expression x 2 + 10 x + 25 . This is a perfect square trinomial, which can be factored as ( x + 5 ) 2 . So, the factored expression is f ( x ) = 3 x 4 ( x + 5 ) 2
Finding the Zeroes and Multiplicities To find the zeroes, we set the factored polynomial equal to zero: 3 x 4 ( x + 5 ) 2 = 0 This equation is satisfied when x 4 = 0 or ( x + 5 ) 2 = 0 .
If x 4 = 0 , then x = 0 . The exponent of the factor x 4 is 4, so the multiplicity of the zero x = 0 is 4. If ( x + 5 ) 2 = 0 , then x + 5 = 0 , so x = − 5 . The exponent of the factor ( x + 5 ) 2 is 2, so the multiplicity of the zero x = − 5 is 2.
Conclusion Therefore, the zeroes of the graph of f ( x ) are -5 with multiplicity 2 and 0 with multiplicity 4.
Examples
Understanding the zeroes and their multiplicities is crucial in various fields, such as physics and engineering, where polynomial functions are used to model real-world phenomena. For instance, in structural engineering, analyzing the roots of a polynomial equation helps determine the stability of a bridge or building. Similarly, in signal processing, understanding the zeroes of a transfer function is essential for designing filters that selectively amplify or attenuate certain frequencies. The multiplicity of a root can indicate the degree of stability or resonance in these systems, providing valuable insights for design and optimization.
