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Interval notation is a shorthand used in mathematics to describe continuous ranges of numbers along the number line. This method efficiently communicates where a particular condition is met, such as where an inequality holds true.

In interval notation, a round bracket, also known as a parenthesis, \((\) or \()\), indicates that the endpoint is not included in the interval, known as an open interval. A square bracket, \([\) or \(]\), indicates that the endpoint is included, known as a closed interval.

For example, the interval \((2, 5]\) includes all numbers greater than 2 and up to and including 5. If we express a solution where a polynomial is positive, we use this notation to clearly list the intervals that satisfy this condition. If the solution includes intervals \((-abla, 2)\) and \((\frac{5}{2}, 4)\), it means the polynomial inequality holds true for all numbers in these ranges, excluding the endpoints 2, \(\frac{5}{2}\), and 4.

The Rational Root Theorem is a valuable tool for finding potential rational solutions, or roots, of a polynomial equation. It suggests that any rational root of the polynomial equation \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0\) must be of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\).

To use this theorem, list all possible values of \(p\) and \(q\), then form all potential fractions \(\frac{p}{q}\). These fractions represent possible rational roots that can be verified by substitution into the polynomial.

In the step-by-step problem, using the Rational Root Theorem helps identify roots like \(x = \frac{5}{2}\), \(x = 2\), and \(x = 4\). These roots are critical in determining the intervals where the polynomial is greater than zero.

Factoring polynomials is the process of expressing a polynomial equation as the product of simpler polynomials. This technique helps solve polynomial equations and inequalities by breaking down complex expressions into sets of simpler factors.

In the case of polynomial inequalities, once the roots are identified using methods like the Rational Root Theorem, they can be used to factor the polynomial entirely. For example, given roots \(x = \frac{5}{2}\), \(x = 2\), and \(x = 4\), the polynomial can be factored as \((x - 2)\left(x - \frac{5}{2}\right)(x - 4)\).

This factorized form directly relates to solving the inequality, as each factor corresponds to a term where the polynomial changes sign, helping to locate the positive or negative intervals through test points.

Synthetic division is a simplified form of polynomial division, particularly useful for dividing a polynomial by a linear divisor of the form \(x - c\). This method is much quicker than long division and is especially handy when identifying or verifying roots of polynomials and their factors.

To perform synthetic division, write down the coefficients of the polynomial. Then, using a suspected root, such as those identified by the Rational Root Theorem, follow the steps to reduce the polynomial to see if the remainder is zero. If it is, then \(x - c\) is indeed a factor of the polynomial.

In the given problem, synthetic division can verify each potential root and assist in completely factoring the polynomial. This, in turn, helps analyze the intervals needed to solve the polynomial inequality efficiently.