Final answer:
The zeros of the function f(x) = -x(3x-2)²(x+9)⁵ are x = 0 with a multiplicity of 1, x = 2/3 with a multiplicity of 2, and x = -9 with a multiplicity of 5.
Step-by-step explanation:
The student asked to identify the zeros and their multiplicity for the function f(x) = -x(3x-2)²(x+9)⁵.
To find the zeros of the function, we set the function equal to zero and solve for x:
- For the term x, the zero is 0 with a multiplicity of 1.
- For the term (3x-2)², setting 3x-2 = 0 gives a zero of 2/3 with a multiplicity of 2.
- For the term (x+9)⁵, setting x+9 = 0 gives a zero of -9 with a multiplicity of 5.
The zeros of the function are x = 0, x = 2/3, and x = -9 with their respective multiplicities of 1, 2, and 5.