at which root does the graph of f(x)=(x - 5)^3(x + 2)^2 touch the x - axis?\n-5\n-2\n2\n5

Explanation:

Step1: Recall the root - multiplicity rule

If a polynomial (y = f(x)) has a factor ((x - a)^n), when (n) is even, the graph of the function touches the (x) - axis at (x=a), and when (n) is odd, the graph of the function crosses the (x) - axis at (x = a).

Step2: Analyze the factors of (f(x)=(x - 5)^3(x + 2)^2)

For the factor ((x - 5)^3), the multiplicity (n = 3) (odd), so the graph of (y=f(x)) crosses the (x) - axis at (x = 5). For the factor ((x + 2)^2), the multiplicity (n=2) (even). So the graph of (y = f(x)) touches the (x) - axis at (x=-2).

Answer:

B. -2