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A cubic equation is a polynomial equation of degree three, which means its highest power of the variable (usually denoted as \(x\)) is three. A typical form of a cubic equation is \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(a eq 0\). In this form:

• \(a\) is the coefficient of \(x^3\)
• \(b\) is the coefficient of \(x^2\)
• \(c\) is the coefficient of \(x\)
• \(d\) is the constant term

Cubic equations can be solved using a variety of methods, including factoring, synthetic division, or even using the cubic formula. Factoring by grouping, as done in the original exercise, is one efficient method when the polynomial can be separated into parts that share a common factor. Solving cubic equations often reveals three roots, which can be real or complex numbers.
Understanding cubic equations is essential as they frequently appear in various areas of mathematics and applied sciences.

The Greatest Common Factor (GCF) is a key concept when simplifying or factoring expressions. The GCF of a set of terms refers to the largest factor that divides each term without leaving a remainder. Common factors can be numbers or variables.

• To find the GCF, identify the highest power of each variable common to all terms.
• For coefficients, find the largest integer that can divide each without a remainder.

For example, in the original exercise, for the group \(12x^3 + 8x^2\), the GCF was identified as \(4x^2\). Finding the GCF is crucial because it allows us to factor expressions fully and simplify them, making further algebraic manipulation easier. In the process of factoring by grouping, once we extract the GCF, the remaining expressions are often simpler to manage or solve.

Solving quadratic equations is a fundamental skill in algebra, key to solving many real-world problems. A quadratic equation is typically in the form \(ax^2 + bx + c = 0\). There are different methods to solve these, including:

• Factoring: If the equation can be factored, it's often the simplest way to find the solution.
• Quadratic Formula: This formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is applicable to all quadratic equations and finds roots directly.
• Completing the Square: This method involves rearranging the equation into a perfect square trinomial, which can be easily solved.

In the original exercise, after factoring the cubic equation, a quadratic of the form \(4x^2 - 1 = 0\) appeared, which was solved by setting it to zero and taking the square root. This approach shows how interconnected algebraic methods can simplify complex problems.