Final answer:
The equation (x - 5)²(x + 2) = 0 has two roots: 5 with a multiplicity of 2, and -2 with a multiplicity of 1.
Therefore, the correct answer is: option : "The root -2 has a multiplicity of 1. The root 5 has a multiplicity of 2.
Step-by-step explanation:
To find the roots, we set each factor equal to zero and solve for x.
Setting (x - 5)² = 0, we find that x = 5.
Since (x - 5) appears squared, the root 5 has a multiplicity of 2.
This means the graph of the equation touches and turns around at the x-axis at this point, rather than crossing it.
Setting (x + 2) = 0, we find that x = -2.
As this factor is not raised to a power higher than 1, the root -2 has a multiplicity of 1, which means the graph of the equation will cross the x-axis at this point.
Therefore, the correct option is: The root -2 has a multiplicity of 1. The root 5 has a multiplicity of 2.