Verified Answer :

  • Apply the Zero Product Property to set each factor to zero: \(x-16=0\), \(x-10=0\), \(x+7=0\), and \(x+17=0\).
  • Solve each equation to find the roots: \(x=16\), \(x=10\), \(x=-7\), and \(x=-17\).
  • Select a positive root from the set, such as 10 or 16.
  • The final answer is \(\boxed{16}\).

Explanation

  1. Applying the Zero Product Property

To find the solution to the given equation \((x - 16)(x - 10)(x + 7)(x + 17) = 0\), we utilize the Zero Product Property. This property states that if the product of multiple factors is equal to zero, then at least one of the factors must be equal to zero. This allows us to break down the complex equation into simpler, linear equations.

  1. Solving for the Roots

We set each individual factor equal to zero:

  1. \(x - 16 = 0 \implies x = 16\)
  2. \(x - 10 = 0 \implies x = 10\)
  3. \(x + 7 = 0 \implies x = -7\)
  4. \(x + 17 = 0 \implies x = -17\)

Thus, the complete set of solutions for the equation is \(x \in \{16, 10, -7, -17\}\).

  1. Identifying Positive Solutions

The question specifically asks for a positive solution. Looking at the roots we found, \(\{16, 10, -7, -17\}\), we can see that the positive values are \(10\) and \(16\). Since the question asks for "a" positive solution, either 10 or 16 is a correct answer. We will provide 16 as the solution.

Examples

The Zero Product Property is used frequently in engineering and physics when solving for the time at which an object reaches a certain position. For instance, if the height of a projectile is described by a factored polynomial equation set to zero, finding the roots allows you to determine the time intervals during which the object is at a specific height, helping to predict landing times or peak heights. Understanding this simple yet powerful property allows for efficient problem solving in various real-world scenarios.

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