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

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.

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