Final answer:
The real zeros of the polynomial f(x) are x = -6 with multiplicity 2, x = 1/3 with multiplicity 1, and x = -2 with multiplicity 5. The value x = 0 is not a zero as it is not a solution to the polynomial equation, and multiplicities reflect how the graph of the function behaves near those zeros.
Step-by-step explanation:
The question asks to find all real zeros of the polynomial function f(x) = (x+6) ²(3x-1) (x+2) ⁵ and note their multiplicities. The zeros of this polynomial function are the values of x that make f(x) = 0. From the given function, we can directly see the zeros by looking at the factors:
- x = -6 (Multiplicity: 2): This zero comes from the factor (x+6) ², indicating that x = -6 is a root with multiplicity 2.
- x = 1/3 (Multiplicity: 1): The factor (3x-1) gives the zero x = 1/3 with multiplicity 1.
- x = -2 (Multiplicity: 5): The factor (x+2) ⁵ indicates that x = -2 is a root with a high multiplicity of 5.
It is also important to note that x = 0 (Multiplicity: 0) is not a zero of the polynomials because it does not appear as a solution when setting each factor equal to zero. The multiplicities indicate how many times each zero is repeated; a higher multiplicity means the graph of the function touches or intersects the x-axis at the corresponding point and turns around for even multiplicities or passes through the axis for odd multiplicities.