Question

At which root does the graph of f(x) = (x – 5)3(x + 2)2 touch the x axis?

Answer 1

The answer for the exercise is: -2
The explanation is shown below:
You have the following function given in the problem above: f(x)=(x-5)³(x+2)². As you can see, (x+2)² is a double root, therefore, it touches the x-axis but it does nos cross it. Therefore, the answer is -2.
When you plot the function, you obtain the graph attached.

Answer 2

we have that
f(x) = (x – 5)³(x + 2)²

we know that
the roots can touch or cross the x axis

The way you can tell if the graph is going to cross the x-axis or just touch the x-axis is by looking at the power of the factor.

(x - 5)³ has a power of 3 which is an ODD number. An ODD power means that the graph will cross through the x-axis.

(x + 2)² has a power of 2 which is an EVEN number. An EVEN power means that the graph will touch the x-axis.

therefore

the answer is
the graph of f(x) touch the x axis for the root x=-2

see the attached figure