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Understanding the roots of a cubic equation is essential for breaking down complex polynomials into more manageable parts. A cubic equation is a polynomial of degree three and it is expressed in the form \(ax^3 + bx^2 + cx + d = 0\), where \(a, b, c,\) and \(d\) are constants, and \(a \eq 0\). These equations are known to have up to three roots, which represent the points where the graph of the polynomial intersects the x-axis.

In our example, we have the cubic equation \(2 x^{3}-3 x^{2} \cos (A-B)-2 x \cos ^{2}(A+B)+\sin 2 A \sin 2 B \cdot \cos (A-B)=0\). To solve for the roots, one can look for specific values that satisfy the equation or simplify the equation further down to a quadratic in cases where there is no constant term (like when \(x=0\)).

It's paramount to remember that finding the roots of a cubic equation sometimes may involve complex numbers, and in this context, complex does not mean difficult, but rather numbers that have a real part and an imaginary part. As such, the roots can be all real or a mix of real and complex numbers depending on the discriminant of the equation.

When faced with a quadratic equation, which typically looks like \(ax^2 + bx + c = 0\), the quadratic formula is an indispensable tool. This formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), and it is derived from completing the square in the quadratic equation. The quadratic formula provides the solutions to the equation and includes the possibility of both real and complex roots depending on the value beneath the square root sign, known as the discriminant \( (b^2 - 4ac) \).

When applying the quadratic formula to a simplified form of a cubic equation, as in our case, we treat the cubic equation as a quadratic one by setting the constant to zero and then solving for \(x\). It allows us to easily find the remaining non-zero roots after identifying one root that made the equation equal to zero.

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. These identities are used to simplify trigonometric expressions or solve trigonometric equations. Common identities include \(\sin^2(\theta) + \cos^2(\theta) = 1\), the angle sum and difference identities, and the double angle formulas like \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\).

In our exercise, we encounter terms like \(\cos(A-B)\) and \(\cos^2(A+B)\), which could be simplified using these identities, especially when determining if \(x=0\) is a root. The analysis of trigonometric expressions plays a critical role to determine if the conditions for \(x=0\) to be a root are met. Considering trigonometric identities can also lead to insights about the periodicity and other characteristics of the roots that may relate to specific angles.

In algebra, roots of an equation can be real, complex, or a combination of both. Real roots can be visualized on a number line, while complex roots consist of a real part and an imaginary part and are often represented in the complex plane. The Fundamental Theorem of Algebra tells us that every non-zero, single-variable, degree \(n\) polynomial with complex coefficients has, counted with multiplicity, exactly \(n\) roots.

The presence of complex roots for cubic equations usually indicates that the discriminant of the equation is negative, but this needs to be verified by actually calculating the discriminant. In our cubic equation example, the sine and cosine functions add complexity and may imply the existence of trigonometric roots, which can be either real or complex. Determining the nature of the roots—whether they're real or complex—is vital for understanding the behavior of the cubic equation within different ranges of values and for graphing the function accurately.