91Ó°ÊÓ

Interval notation is a concise way of representing subsets of the real number line. It involves using brackets and parentheses to indicate whether endpoints are included in the interval (brackets) or not (parentheses). To solve our polynomial inequality, we need to understand how to express the solution set. For instance, \[x, y\] denotes all numbers between x and y, including x and y. Conversely, \(x, y\) refers to all numbers between x and y, excluding x and y themselves. When dealing with infinity, since infinity is not a number but a concept, we always use parentheses, as in \(x, \infty\). Our final solution in interval notation is \[[-2,2] \cup [2,\infty)\], which combines two intervals where the polynomial fulfills the inequality condition.

Graphing solution sets provides a visual understanding of where a polynomial's value meets a specified condition on the real number line. To graph the solution set of our polynomial inequality, we pinpoint the roots of the polynomial on the number line and test intervals to determine where the polynomial meets the inequality condition. We use solid dots on the roots -2 and 2 to indicate that they are part of the solution set (since the inequality is \(\geq 0\)). The intervals where the polynomial tested positive or zero are then shaded or highlighted to depict the solution set graphically. In this case, the number line would show the range \[[-2,2]\] as a solid line, continuing past 2 towards infinity.

The real number line is a straight, horizontal line where every point corresponds to a unique real number and it serves as the canvas for graphing solution sets. It can be helpful when visualizing solutions to inequalities. On this line, points to the right represent larger numbers, and points to the left represent smaller numbers. The real number line can represent all kinds of numbers: rational, irrational, integers, and real numbers. When solving inequalities like the given polynomial inequality, we use the real number line to understand which intervals include numbers that make the inequality true.

Factoring polynomials is essential in solving polynomial inequalities. This process involves breaking down a polynomial into simpler 'factor' polynomials whose product is the original polynomial. When factored, we can easily find the zeros or roots of the polynomial, which are the x-values that make the polynomial equal to zero. These zeros are crucial for establishing intervals for testing in our inequality. In our example, the polynomial \(x^{3}+2 x^{2}-4 x-8\) can be factored into \[ (x+2)(x-2)(x+2) = 0\], revealing the roots; \(x=-2\) and \(x=2\). By identifying these roots, we can investigate the behavior of the polynomial within the created intervals.