To identify at which root the graph of the given function touches the x-axis, we need to understand how the roots of a polynomial relate to its graph.
The polynomial in question is:
\[ f(x) = (x-5)^3 (x+2)^2 \]
In this function, we have two roots: \( x = 5 \) and \( x = -2 \). These roots come from the factors of the polynomial being set to zero.
However, this function doesn't just have simple roots; each root has a multiplicity associated with it, which comes from the exponent on the factor. The root \( x = 5 \) comes from the factor \( (x-5) \), which is raised to the power of 3, making it a root with a multiplicity of 3. Multiplicities tell us how the graph will behave at those roots.
Roots with an odd multiplicity will cross the x-axis. This means the graph goes down to touch the axis and then continues on the other side (either turns up if approaching from below, or turns down if approaching from above).
Roots with an even multiplicity will touch the x-axis and turn back without crossing it. The graph will approach the axis, just touch it, and then reverse direction as if it were reflecting off the axis.
Given that \( (x-5)^3 \) has an odd exponent (multiplicity of 3), the graph will cross the x-axis at \( x = 5 \).
On the other hand, \( (x+2)^2 \) has an even exponent (multiplicity of 2). Therefore, at \( x = -2 \), the graph of the function will merely touch the x-axis and not cross it.
So, the correct answer to the question "At which root does the graph of \( f(x) = (x-5)^3(x + 2)^2 \) touch the x-axis?" is \( x = -2 \).
