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Factoring polynomials is akin to breaking down a number into its multiplicative components, but instead of numbers, we deal with expressions involving variables. This step is pivotal in solving equations as it simplifies complex problems into manageable pieces. To understand it, consider the cubic equation from our exercise, \(u^{3}-3u=0\).

We begin by looking for common factors in each term. Notice how each term contains the variable \(u\). We then 'take out' this common factor, which in mathematical terms means we apply the distributive property in reverse, rewriting the original equation as \(u*(u^2-3) = 0\). This is now a product of two simpler expressions, and it sets the stage for employing the zero product property for further solving the equation. Polynomials can often be factored in this way, especially when there's a clear common factor, or when a pattern, like a difference of squares or sum of cubes, is present.

The zero product property is a crucial concept in algebra and serves as a trusty tool in our mathematical toolkit. Simply put, it tells us that if the product of two or more factors equals zero, at least one of the factors must also be equal to zero. This principle guides us from a product of factors to a set of simpler equations.

In our exercise, after factoring the polynomial, we utilized the zero product property. We set each factor equal to zero individually because if their product is zero, one or both of them has to be zero. We obtain two separate equations: \(u = 0\) and \(u^2 - 3 = 0\). This step is like breaking a chain where each link must be tested to find the weak one that breaks the entire chain. By solving these simpler equations, the solutions to the original problem become clearer and more accessible.

The square root is a mathematical operation that asks 'what number, when multiplied by itself, gives us the original number?' For example, the square root of 9 is 3, because \(3 \times 3 = 9\). Applying this concept to our exercise helps us solve the equation \(u^2 - 3 = 0\).

First, we isolate the squared term by adding 3 to each side, resulting in \(u^2 = 3\). To find the value of \(u\), we take the square root of both sides of the equation. However, it's crucial to recognize that there are always two solutions for a squared variable: a positive and a negative square root. Therefore, the equation \(u^2 = 3\) yields two solutions: \(u = \(\sqrt{3}\)\) and \(u = -\sqrt{3}\). Why? Because both \((\(\sqrt{3}\))^2\) and \((-\sqrt{3})^2\) equal 3. In the context of the exercise, this means that we have found the two remaining solutions besides \(u = 0\), completing the set of solutions for the original cubic equation.