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Understanding interval notation is crucial when presenting solutions to inequalities in calculus. It's a simplified way to express a range of numbers without listing every individual point. You often see it written with parentheses like this:

• (a, b): Includes all numbers between a and b but not a and b themselves. It represents an open interval.
• [a, b]: Includes the numbers a and b along with all numbers between them. This is a closed interval.
• (a, b] or [a, b): Represents half-open intervals. The number with the bracket is included in the set.

In our example, the solution set is indicated as \((-1, 1) \cup (1, \infty)\). The cup symbol \((\cup)\) represents a union of intervals, meaning individual sections combined as part of the solution.
This means the graph is shaded between -1 and 1, not including the endpoints, and similarly extends from 1 to positive infinity.

Polynomial factoring is the method of breaking down a polynomial into simpler polynomial factors that, when multiplied, yield the original polynomial. This process can make solving inequalities and other algebraic problems more manageable. In the given problem, we employ factoring to find the roots or zeros of the polynomial, which helps identify the critical points.
To factor the given polynomial \(x^{3}-x^{2}-x+1\), we begin by grouping terms: \(x^3 - x^2 - x + 1 = (x^2 - 1)(x - 1)\). Then, you'll observe it further simplifies to \((x - 1)^2(x + 1)\).
This factored form provides critical insights, specifically the zeros or points where the expression equals zero. It provides us with the critical points to test for intervals where the original inequality remains valid.

Critical points are values of \(x\) where the function's derivative is zero or undefined, or in this context, where our factored polynomial equals zero. These points help divide the number line into intervals for testing whether the inequality is true or false.
In the exercise, factoring \((x - 1)^2(x + 1)\) led us to discover these critical points:

• \(x = 1\): Obtained from solving \((x - 1)^2 = 0\).
• \(x = -1\): Obtained from solving \((x + 1) = 0\).

Once critical points are found, they divide the number line into segments or intervals. For each interval, we "test" with a number to determine whether it satisfies the inequality.
Here, choosing values like \(x = -2\) in \((\infty, -1)\), \(x = 0\) in \((-1, 1)\), and \(x = 2\) in \((1, \infty)\), helps validate which parts of the intervals satisfy the inequality. This helps ensure a complete solution expressed in interval notation.