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The zero product rule states that if a product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. Mathematically, if \(ab = 0\), then either \(a = 0\) or \(b = 0\) or both.

We will use a proof by contradiction to prove the zero product rule. Suppose that neither \(a\) nor \(b\) is equal to zero. When we multiply the two factors, \(ab\), the product is nonzero, given that the product of two nonzero numbers is nonzero. But this contradicts our initial assumption that \(ab = 0\). So, our assumption that neither of the factors equal to zero is false, thus we conclude that at least one of the factors must be zero.

Let's take an example of a quadratic equation: \(x^2 - 5x = 0\). First, we need to factor this equation to apply the zero product rule. Factor the equation: \(x(x - 5) = 0\) Now we can apply the zero product rule: Either \(x = 0\) or \(x - 5 = 0\) Thus, the solutions to the quadratic equation are \(x = 0\) and \(x = 5\).

The zero product rule can also be applied to higher-degree polynomial equations. For example, let's consider a cubic polynomial equation: \(2x^3 - 3x^2 + x = 0\). Factor the equation: \(x(2x^2 - 3x + 1) = 0\) Now we can apply the zero product rule: Either \(x = 0\) or \(2x^2 - 3x + 1 = 0\) In this case, the higher-degree polynomial (quadratic) will need further factoring or other solving methods (e.g., quadratic formula) to find the remaining solutions. In summary, the zero product rule is a useful tool for solving polynomial equations, as it allows us to determine the solutions by setting each factor equal to zero. However, depending on the complexity of the equation, further factoring or solving methods may be required.