All roots either touch or cross the x-axis.
Look carefully at the given function f(x) = (x – 5)^3*(x + 2)^2.
Notice how the root x=5 occurs 3 times. We say that this root has a "multiplicity" of 3, which is an odd number. The graph will cross the x-axis at x=5. There's a "horizontal point of inflection" at x= 5.
x= -2, the other root, has a multiplicity of 2, which is even. The graph will touch the x-axis at x = -2, but not cross it.
Check this out by graphing f(x) = (x – 5)^3*(x + 2)^2
on your calculator.
Please use " ^ " to indicate exponentiation.