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Factoring polynomials is a crucial skill when solving polynomial inequalities. It is the process of breaking down a complex expression into simpler terms that, when multiplied together, yield the original polynomial. For instance, consider the polynomial inequality from the exercise,

\( 3x^{2}-5x \leq 0 \).

The first step in factoring is to identify any common factors. Here, we can see that 'x' is common to both terms. Factoring 'x' out, the expression becomes:

\( x(3x - 5) \leq 0 \).

This representation is much easier to work with because it simplifies the problem to finding the zeros of the polynomial. These zeros are the 'roots' of the inequality, which serve as critical points that help in determining the intervals where the inequality will be satisfied. In this example, setting each factor equal to zero gives us the roots \( x = 0 \) and \( x = \frac{5}{3} \). The factorization step is preparatory and sets the stage for further analysis in solving inequalities.

Interval notation is a concise way of writing sets of numbers, often used in the context of solutions for inequalities. It's essential to become familiar with this notation because it simplifies the communication of solution sets for inequalities and calculus problems.

For example, using interval notation, the set of all real numbers less than 5 is written as \( (-\infty, 5) \), which is read as 'all numbers from negative infinity up to, but not including, 5'. If 5 were to be included in the set, the interval would be written as \( (-\infty, 5] \). The square bracket indicates inclusion in the set, while the parentheses indicate exclusion.

In the polynomial inequality \( 3x^{2}-5x \leq 0 \), the solution set includes the numbers where the polynomial is less than or equal to zero. Including the roots, as they make the polynomial exactly zero, the solution is represented as \( (-\infty, 0] \cup [\frac{5}{3}, \infty) \). The union symbol \( \cup \) combines the two intervals, clearly showing that the solution includes all numbers up to and including 0 and all numbers from \( \frac{5}{3} \) onward.

Graphing the solution sets on a number line visually represents where the solutions to an inequality lie. After factoring the polynomial and finding the roots, we mark these critical points on the number line. It's helpful because it illustrates the intervals over which the original inequality holds true.

In our exercise, after factoring and finding the critical roots, we've determined the solution set to be \( (-\infty, 0] \cup [\frac{5}{3}, \infty) \). On a number line, each point is marked, and we then decide which sections to shade based on the inequality. In this case, we want to shade all the sections where the polynomial is less than or equal to zero. Including the roots in the shading indicates that the points \( 0 \) and \( \frac{5}{3} \) are part of the solution set as they satisfy the inequality \( 3x^{2}-5x \leq 0 \) exactly. This graphing technique is a potent tool as it provides an instant visualization of all possible solutions to an inequality.