- Set the function \(f(x) = -4x(x^2 - 4)(x^2 + 16)\) to zero and apply the Zero Product Property.
- Solve each factor individually: \(-4x = 0\), \(x^2 - 4 = 0\), and \(x^2 + 16 = 0\).
- Identify that \(-4x = 0\) gives \(x = 0\), and \(x^2 - 4 = 0\) yields \(x = \pm 2\), while \(x^2 + 16 = 0\) has no real roots.
- Conclude that the real zeros of the function are \(\boxed{-2, 0, 2}\).
Explanation
- Analyzing the Function Factors
The function is given by \(f(x) = -4x(x^2 - 4)(x^2 + 16)\). To find the zeros of the function, we must determine all values of \(x\) that satisfy the equation \(f(x) = 0\). According to the Zero Product Property, a product of factors is equal to zero if and only if at least one of its factors is zero. Thus, we set each factor equal to zero:
- \(-4x = 0\)
- \(x^2 - 4 = 0\)
- \(x^2 + 16 = 0\)
We will analyze these factors to find all real solutions for \(x\).
- Solving for Zeros of Each Factor
Now, let's solve each factor equation for \(x\):
- For the factor \(-4x = 0\), dividing both sides by \(-4\) gives \(x = 0\). Since 0 is a real number, this is one of the zeros.
- For the factor \(x^2 - 4 = 0\), we can add 4 to both sides to get \(x^2 = 4\). Taking the square root of both sides, we find \(x = \pm \sqrt{4}\), which gives \(x = 2\) and \(x = -2\). Both 2 and -2 are real numbers, so these are also zeros.
- For the factor \(x^2 + 16 = 0\), we subtract 16 from both sides to get \(x^2 = -16\). For any real number \(x\), the square \(x^2\) is always non-negative (\(x^2 \ge 0\)). Therefore, there is no real number \(x\) such that \(x^2 = -16\). This factor provides no real zeros.
- Conclusion and Final Result
Collecting the results from the previous step, the real zeros of the function \(f(x)\) are \(0\), \(2\), and \(-2\). Therefore, the set of all real zeros is \(\{-2, 0, 2\}\). We can conclude that these are the only values for which the given function equals zero. The final answer is \(\boxed{-2, 0, 2}\).
Examples
In physics, finding the zeros of a function is crucial for determining equilibrium points or moments when an object crosses a specific path. For example, if the trajectory of a projectile is modeled by a function \(h(t)\) representing height over time, the zeros of \(h(t)\) tell us the exact times when the projectile hits the ground. Understanding how to factor polynomials and solve for their roots allows us to predict these critical moments precisely in engineering, navigation, and many other real-world applications.