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

The roots are x = 5 and x = −2. A graph touches the x-axis at a root with an even exponent, because the function stays on the same side of the axis there. Since (x + 2) has exponent 2, the graph touches at x = −2; at x = 5, the exponent 3 is odd, so the graph crosses.

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