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Factoring polynomials is a crucial step in simplifying rational functions and identifying other characteristics of their graphs. When you encounter a polynomial, the first task is to look for common factors among its terms. For example, consider the polynomial expression \(3x^2 - 3x - 6\). Here, the common factor is \(3\). By factoring this number out, the expression reduces to \(3(x^2 - x - 2)\).

Once the common factor is extracted, further factor the polynomial \(x^2 - x - 2\). This involves finding two numbers whose product is \(-2\) and whose sum is \(-1\). The appropriate choice for these numbers is \(-2\) and \(1\), which leads to the factorization \((x - 2)(x + 1)\).

• Extract any common numeric or variable factors.
• Break down the resulting polynomial further into binomials, if possible.
• Use techniques such as grouping or quadratic equivalents.

Factoring is not just an exercise in simplification—it reveals solutions and helps identify potential issues like undefined points or other characteristics in graphs.

The domain of a function is the set of all possible inputs (x-values) for the function that produce valid outputs. For rational functions, such as \(r(x) = \frac{3x^2 - 3x - 6}{x - 2}\), the domain excludes any value of \(x\) that makes the denominator zero, as division by zero is undefined.

In the case of \(r(x)\), the denominator \(x - 2\) means \(x = 2\) is excluded from the domain. Thus, the domain of \(r(x)\) is all real numbers except \(x = 2\). For graphing purposes, these excluded values can produce holes, and understanding them is crucial for constructing an accurate graph of the function.

• Identify the values that make the denominator zero.
• Exclude these values from the domain of the function.
• Remember to adjust the graph accordingly to reflect these missing points.

The domain helps us understand the scope of the function and ensures that we are aware of any discontinuities to expect.

Graphing functions involves translating the mathematical expression of a function into visual form on a coordinate plane. For rational functions, this can involve identifying and plotting any simplified linear or nonlinear graphs, accounting for modifications due to excluded domains.

For example, if we simplify \(r(x) = \frac{3(x-2)(x+1)}{x-2}\) to \(3(x+1)\), it initially appears as the line \(y = 3x + 3\). But, since \(x = 2\) is not in the domain, the graph will have a disruption at \((2, 9)\).

• Identify the core graph structure after simplification.
• Locate and mark any holes or asymptotes on the graph.
• Ensure to indicate and adjust lines based on these disrupted points.

Graphing is about capturing both the general shape and specific peculiarities, like missing points or extreme values, to convey the behavior of the function comprehensively.

Holes occur in graphs of rational functions at the x-values that are excluded from the function's domain due to the zeroes in their denominators. Essentially, they are points where the function is not defined, but near which the graph behaves predictably.

Using \(r(x)\) once simplified to \(3(x+1)\), and excluding \(x = 2\), means the point \((2, 9)\) is not on the graph of the simplified linear equation. However, as \(x\) approaches \(2\) from either direction, the y-values approach \(9\). Therefore, a hole is depicted by a small open circle on the graph to show the missing point.

• Identify the point(s) where the original rational function is undefined.
• Mark these points on the graph with open circles to indicate missing data.
• Ensure clarity by showing how close the function value gets at nearby points.

Holes give a gentle nudge that simplicity may hide omissions, urging you to remember that the algebraic process could simplify the appearance but lose the complete story.