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Real zeros are the values of \(x\) for which a function \(f(x)\) equals zero. They are visually represented in a graph as the points where the graph crosses or touches the x-axis. Finding the real zeros of a function is equivalent to solving the equation \(f(x) = 0\).
For polynomial functions, the zeros can be deduced by setting the polynomial equal to zero and solving for \(x\). Each zero represents an x-value that makes the function output zero. In our example, the function \(g(x) = \frac{1}{5}(x + 1)^2(x - 3)(2x - 9)\) has its zeros at the points where \(x\) makes any of the factors equal zero.
These zeros are approximately \(x = -1, 1.5, 3\). These positions correspond to the factors \((x + 1)\), \((x - 3)\), and \((2x - 9)\) becoming zero, respectively. By understanding real zeros, one can gain useful insights into the behavior and characteristics of the function.

A graphing utility is a powerful tool for visualizing mathematical functions. It provides a means to graph equations and functions accurately, allowing us to observe their behavior graphically.
When graphing the function \(g(x) = \frac{1}{5}(x + 1)^2(x - 3)(2x - 9)\), the graphing utility helps identify the zeros by displaying where the curve intersects the x-axis. This intersection is a visual representation of where the function evaluates to zero.
Most graphing utilities also come equipped with specific features to find zeros more precisely. A zero or root feature automates the process of pinpointing exact x-values or provides extremely close approximations where the function crosses or touches the x-axis. By using this functionality, we can identify the zeros as \(x = -1, 1.5, 3\) efficiently.

Multiplicity refers to the number of times a specific zero appears as a solution to a function. In graphical terms, it determines how the graph behaves at the zero.
If a zero has an odd multiplicity, the graph crosses the x-axis at that zero. Conversely, if the multiplicity is even, the graph merely touches the x-axis and then turns back.
Taking the function \(g(x) = \frac{1}{5}(x + 1)^2(x - 3)(2x - 9)\) as an example:

• The zero at \(x = -1\) arises from the factor \((x + 1)^2\) and has multiplicity 2, suggesting the graph touches and then bounces off the x-axis here.
• The zero at \(x = 3\) comes from \((x - 3)\) with a multiplicity of 1, indicating the graph crosses the x-axis.
• Similarly, \(x = 1.5\) is derived from \((2x - 9)\) with a multiplicity of 1, meaning the graph also crosses at this point.

The concept of multiplicity helps us predict and understand the shape and direction of a function’s graph near each zero.