What are the zeros of the function f(x) = 5x³ + 5x² – 30x and what are their multiplicities?

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Finding the Zeros of the Function

The function you provided is:
f(x) = 5x³ + 5x² – 30x

To find the zeros of this function, we need to solve for x when f(x) = 0:

5x³ + 5x² - 30x = 0

We can factor out the common term:

5x(x² + x - 6) = 0

This gives us one obvious zero:

  • x = 0

Next, we need to factor the quadratic equation x² + x – 6. We can do this by looking for two numbers that multiply to -6 and add to 1 (the coefficient of x):

(x + 3)(x - 2) = 0

So, we can further rewrite the function with its factored form:

5x(x + 3)(x - 2) = 0

This gives us three zeros:

  • x = 0
  • x = -3
  • x = 2

Determining the Multiplicities

Now, let’s analyze the multiplicity of each zero:

  • x = 0: This factor appears only once in the factored form, which means its multiplicity is 1.
  • x = -3: This root also appears only once, so its multiplicity is 1.
  • x = 2: Similarly, this root appears only once as well, giving it a multiplicity of 1.

Summary

In summary, the zeros of the function f(x) = 5x³ + 5x² – 30x are:

  • x = 0 (multiplicity: 1)
  • x = -3 (multiplicity: 1)
  • x = 2 (multiplicity: 1)

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