What are the zeros of the polynomial function f(x) = x³ + x² – 20x?

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To find the zeros of the polynomial function f(x) = x³ + x² – 20x, we need to solve the equation f(x) = 0. This means we want to find the values of x for which the function equals zero.

First, we start by factoring the polynomial. We can rearrange the function as follows:

f(x) = x(x² + x – 20)

Next, we can see that one factor is x, which gives us the first zero:

x = 0

Now, we need to solve the quadratic equation x² + x – 20 = 0. We can do this by using the quadratic formula:

x = (-b ± √(b² – 4ac)) / 2a

In this case, a = 1, b = 1, and c = -20. Plugging these values into the quadratic formula gives us:

x = (-1 ± √(1² – 4(1)(-20))) / (2(1))

This simplifies to:

x = (-1 ± √(1 + 80)) / 2

x = (-1 ± √81) / 2

x = (-1 ± 9) / 2

Now we have two solutions:

  • x = (8) / 2 = 4
  • x = (-10) / 2 = -5

Therefore, we find that the zeros of the polynomial function f(x) = x³ + x² – 20x are:

  • x = 0
  • x = 4
  • x = -5

In conclusion, the complete set of zeros for the polynomial is 0, 4, and -5.

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