Verified Answer :

  • Recognize the function as a product of factors: \(x, (x-2),\) and \((x+6)^2\).
  • Apply the zero-product property: set each factor equal to zero: \(x=0\), \(x-2=0\), and \((x+6)^2=0\).
  • Solve each resulting equation: \(x=0\), \(x=2\), and \(x=-6\).
  • Conclude that the zeros are \(-6, 0, \text{ and } 2\), which corresponds to the first option: \(\boxed{-6, 0, 2}\).

Explanation

  1. Analyzing the Function and the Zero-Product Property

To find the zeros of the function \(y = x(x - 2)(x + 6)^2\), we need to determine the values of \(x\) for which \(y = 0\). According to the zero-product property, a product of factors is zero if and only if at least one of the factors is equal to zero. Therefore, we set each individual factor to zero and solve for \(x\).

  1. Finding the First Zero

The first factor is \(x\). Setting this to zero gives:
\[x = 0\]
This provides our first zero.

  1. Finding the Second Zero

The second factor is \(x - 2\). Setting this to zero gives:
\[x - 2 = 0\]
Adding \(2\) to both sides, we find:
\[x = 2\]
This provides our second zero.

  1. Finding the Third Zero

The third factor is \((x + 6)^2\). Setting this to zero gives:
\[(x + 6)^2 = 0\]
Taking the square root of both sides, we get:
\[x + 6 = 0\]
Subtracting \(6\) from both sides, we find:
\[x = -6\]
This provides our third zero.

  1. Conclusion and Final Result

Combining these results, the zeros of the function \(y = x(x - 2)(x + 6)^2\) are \(x = -6, 0, 2\). Comparing this result with the given options, we select the correct one.

  • \(-6, 0\), and \(2\)
  • \(-6, 0, 2\), and \(6\)
  • \(-2, 0\), and \(6\)
  • \(-6\) and \(2\)

The correct option is \(-6, 0, \text{ and } 2\). The final answer is \(\boxed{-6, 0, 2}\).

Examples

Understanding how to find the zeros of a function is crucial in fields like physics and engineering, where zeros often represent physical thresholds, such as the point where a projectile hits the ground or where the power output of a system is zero. For example, in bridge design, determining where an equation describing the bridge's curve equals zero helps engineers identify the points where the structure meets its supports, ensuring stability and safety. By mastering this concept, you are learning to solve for critical points that define how systems behave in the real world.