What are the zeros of the polynomial function f(x) = x^3 + 10x^2 + 24x?

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Finding the Zeros of the Polynomial f(x) = x³ + 10x² + 24x

To find the zeros of the polynomial function f(x) = x³ + 10x² + 24x, we need to determine the values of x that make the function equal to zero.

Step 1: Factor the Polynomial

The first step in finding the zeros is to factor the polynomial. Noticing that each term of the polynomial shares a common factor of x, we can factor it out:

f(x) = x(x² + 10x + 24)

Now, we need to factor the quadratic expression x² + 10x + 24.

Step 2: Factoring the Quadratic

To factor the quadratic x² + 10x + 24, we look for two numbers that multiply to 24 (the constant term) and add up to 10 (the coefficient of the middle term). The numbers 4 and 6 work since:

  • 4 × 6 = 24
  • 4 + 6 = 10

Thus, we can rewrite the quadratic as:

x² + 10x + 24 = (x + 4)(x + 6)

Incorporating this back into our factored expression gives us:

f(x) = x(x + 4)(x + 6)

Step 3: Setting the Factors to Zero

Now we set each factor equal to zero to find the values of x that make the entire function zero:

  • x = 0
  • x + 4 = 0 ⟹ x = -4
  • x + 6 = 0 ⟹ x = -6

Conclusion

Therefore, the zeros of the polynomial function f(x) = x³ + 10x² + 24x are:

  • x = 0
  • x = -4
  • x = -6

These are the points where the graph of the polynomial intersects the x-axis.

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