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

-5, -2, 2, 5

**Solution:**

Given: Function f(x) is (x - 5)^{3}(x + 2)^{2}

If a curve touches the x-axis then f(x) = 0

⇒ (x - 5)^{3}(x + 2)^{2} = 0.

But if ab = 0 ⇒ either a = 0 or b = 0 or both zero.

⇒ (x - 5)^{3} = 0 and (x + 2)^{2} = 0

⇒ (x - 5) = 0 and x + 2 = 0

⇒ x = 5 and x = - 2.

These points can be verified from the below graph.

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

**Summary:**

The root of the graph of f(x) = (x - 5)^{3}(x + 2)^{2} touches the x axis at -2 and 5.

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