91Ó°ÊÓ

A quadratic equation is an equation that involves a variable raised to the second power. It can be written in the general form:

• \(ax^2 + bx + c = 0\).

In this exercise, the quadratic equation is \(x^2 - 2x - 3 = 0\). This is obtained from the factorization of the original cubic equation. Quadratic equations can usually be solved by:

• Factoring, if the equation is easily factorable.
• Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the square, another algebraic method.

• In our example, we used factoring: \((x - 3)(x + 1) = 0\), leading us to solutions \(x = 3\) and \(x = -1\).

These solutions are integral-based, as shown by the calculation steps. It's crucial to remember these roots are only part of the solution when considered in a modular context.

Factoring means breaking down an equation into simpler parts, called factors, that when multiplied give the original equation. Some common factoring methods include:

• Finding common factors.
• Using special formulas for recognizing patterns, such as difference of squares.
• Grouping terms.

For this problem, we start with the cubic equation: \(x^3 - 2x^2 - 3x = 0\).

• The first step is taking out the greatest common factor, here \(x\), which transforms the equation into \(x(x^2 - 2x - 3) = 0\) . This shows us one possible solution is \(x = 0\), immediately simplifying the complexity of the polynomial.
• The quadratic part \(x^2 - 2x - 3\) was further factored into \((x - 3)(x + 1) = 0\). Each factor gives us additional possible solutions when solved individually in the modular context.

Factoring simplifies equations and is a critical step in solving polynomial equations efficiently.

To find solutions in \( \mathbb{Z}_n \), specifically \( \mathbb{Z}_{12} \) for this exercise, we must find values of \(x\) that satisfy the equation when considered under modulo 12 arithmetic. Here are some key points:

• \( \mathbb{Z}_n \) represents the set of integers modulo \(n\), which includes numbers from 0 to \(n - 1\).
• For \(x^3 - 2x^2 - 3x \equiv 0 \pmod{12}\), we test possible integer values, \(x = 0, 1, \ldots, 11\).
• After solving the quadratic in previous steps, potential solutions include \(x = 0, 3,\) and presume \(x = -1\), which translates to \(x = 11\) in \(\mathbb{Z}_{12}\).

Each of these values, when substituted back into the original equation, must result in an equivalence that holds true, ensuring these are correct solutions within this modular system.

The term 'modulo' represents the remainder operation in mathematics. It’s a crucial part of modular arithmetic, which deals with integers and their remainders when divided by a number \(n\). This operation is symbolized by \(%\) or \(\pmod{n}\). Now, what does this mean in practice? When we say that \(a \equiv b \pmod{n}\), it means that \(a\) and \(b\) leave the same remainder when divided by \(n\).

• For instance, \(1056 \equiv 0 \pmod{12}\), since 1056 divided by 12 leaves no remainder.

When working through the problem, verify each potential solution, for example \(x = 0, 3, 11\), to ensure that they satisfy \(x^3 - 2x^2 - 3x \equiv 0 \pmod{12}\). This alignment confirms their legitimacy as solutions in \( \mathbb{Z}_{12} \).The modulo operation simplifies and organizes arithmetic into a manageable structure, allowing us to solve complex equations within finite integer systems.