Question

Determine the total number of roots of each polynomial function using the factored form.

f(x)=(x-6)^2(x+2)^2

Answer 1

The polynomial function f(x) = (x - 6)^2 (x + 2)^2 has four roots.

The factor (x - 6)^2 has a root of 6 with a multiplicity of 2, which means that the graph of f(x) touches the x-axis at x = 6 but does not cross it. The factor (x + 2)^2 has a root of -2 with a multiplicity of 2, which means that the graph of f(x) touches the x-axis at x = -2 but does not cross it.

Since each factor has a multiplicity of 2, the graph of f(x) touches the x-axis at each root but does not cross it. Therefore, the total number of roots of the polynomial function f(x) = (x - 6)^2 (x + 2)^2 is 4.