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To solve equations like \(5(x-3)(x-7)=0\), we rely on the Zero Product Property. This property is crucial in algebra and states that if the product of multiple factors equals zero, at least one of those factors must be zero. This is because zero multiplied by any number is still zero. For example, if \(a \times b = 0\), then either \(a = 0\) or \(b = 0\), or both.

Applying this to our exercise, we have three factors: \5, (x-3)\, and \ (x-7)\. The number 5 is a non-zero constant and does not affect the equation. Hence, we focus on the other two factors that could potentially be zero to satisfy the Zero Product Property.

Factoring is a method used to simplify and solve polynomial equations by expressing them as a product of their factors. In our given problem, \(5(x-3)(x-7)=0\), the expression is already factored for us. This means we have isolated the important parts of the equation, making it easier to solve.

Here, \(x-3\) and \(x-7\) are two factors of the quadratic equation. To solve this, we treat each factor separately, based on the Zero Product Property. This step of recognizing and handling each factor is the core of the factoring process.

When an equation is not factored, we must use different techniques like grouping, using the quadratic formula, or redistributing terms to break it apart into more manageable pieces.

The roots of an equation are its solutions—the values that make the equation true. In simple terms, these are the values of x that satisfy the equation. For our problem, \(5(x-3)(x-7)=0\), we find the roots by setting each factor equal to zero and solving for x.

We set \(x-3=0\), giving us \(x=3\).
We also set \(x-7=0\), giving us \(x=7\).

Hence, the roots of the equation are \(x=3\) and \(x=7\). These are the solutions that satisfy the original equation. Knowing how to find roots is essential as it helps us understand where the polynomial intersects the x-axis on a graph, offering visual insights and proofs into the nature of the equation.