Find the zeros of p(x) = 5x^6(x+2)^7(x-3)^7.
Also state the multiplicity of each zero of p(x).
A) 0 (multiplicity 6), 2 (multiplicity 7), -3 (multiplicity 7)
B) 2 (multiplicity 7), -3 (multiplicity 7)
C) -2 (multiplicity 7), 3 (multiplicity 7)
D) 0 (multiplicity 6), -2 (multiplicity 7), 3 (multiplicity 7)
Answer (2)
Find the zeros by setting each factor to zero: 5 x 6 = 0 , ( x + 2 ) 7 = 0 , ( x − 3 ) 7 = 0 .
Solve for x to find the zeros: x = 0 , x = − 2 , x = 3 .
Identify the multiplicity of each zero from the exponent of its corresponding factor.
State the zeros and their multiplicities: 0 (multiplicity 6 ), − 2 (multiplicity 7 ), 3 (multiplicity 7 ).
Explanation
Understanding the polynomial We are given the polynomial p ( x ) = 5 x 6 ( x + 2 ) 7 ( x − 3 ) 7 and we want to find its zeros and their multiplicities.
Finding the zeros To find the zeros of p ( x ) , we need to solve the equation p ( x ) = 0 . This means we need to find the values of x that make the polynomial equal to zero.
Solving for x We have 5 x 6 ( x + 2 ) 7 ( x − 3 ) 7 = 0 . This equation is satisfied if any of the factors are equal to zero. So we have:
5 x 6 = 0 ⇒ x 6 = 0 ⇒ x = 0
( x + 2 ) 7 = 0 ⇒ x + 2 = 0 ⇒ x = − 2
( x − 3 ) 7 = 0 ⇒ x − 3 = 0 ⇒ x = 3
Finding the multiplicities The zeros of p ( x ) are 0 , − 2 , and 3 . Now we need to find the multiplicity of each zero. The multiplicity of a zero is the exponent of the corresponding factor in the polynomial.
Stating the multiplicities
For x = 0 , the factor is x 6 , so the multiplicity is 6 .
For x = − 2 , the factor is ( x + 2 ) 7 , so the multiplicity is 7 .
For x = 3 , the factor is ( x − 3 ) 7 , so the multiplicity is 7 .
Final Answer Therefore, the zeros of p ( x ) are 0 (multiplicity 6 ), − 2 (multiplicity 7 ), and 3 (multiplicity 7 ).
Examples
Understanding the zeros and multiplicities of polynomials is crucial in many areas of mathematics and engineering. For example, in control systems, the zeros of a transfer function determine the stability and response characteristics of the system. In signal processing, the zeros of a filter's transfer function determine the frequencies that are attenuated by the filter. In computer graphics, polynomials are used to model curves and surfaces, and their zeros determine the points where the curve or surface intersects a given plane. Knowing the zeros and their multiplicities helps engineers design stable and efficient systems, process signals effectively, and create realistic computer graphics.
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). The identified zeros indicate how the polynomial behaves at those points. It is important to note that the multiple choice responses did not correctly encapsulate the complete solution.
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