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Rational functions can have zeros, which are the values of the variable that make the function equal to zero. To find the zeros, we need to focus on the numerator of the function. In our example, the rational function is given as:\[g(x)=\frac{x^{2}-9}{x+3}\]We set the numerator, \(x^2 - 9\), equal to zero and solve for \(x\). This helps us identify the potential zeros. For our function:- The equation becomes \(x^2 - 9 = 0\).- Solving this gives us \(x = 3\) and \(x = -3\).These solutions suggest that the zeros of this function could be \(x = 3\) and \(x = -3\). However, there's more to finding the true zeros of a rational function. We must examine both the numerator and the denominator, as the denominator can impact the function's behavior.

Factorization is the process of breaking down complex expressions into simpler components called factors. It is particularly useful in identifying zeros of polynomial expressions. For our function, factorization is used to simplify the numerator, \(x^2 - 9\), which is a difference of squares.For any expression that's a difference of squares, there is a formula:\[a^2 - b^2 = (a+b)(a-b)\]Applying this formula to \(x^2 - 9\), where \(a = x\) and \(b = 3\), we get:\[x^2 - 9 = (x + 3)(x - 3)\]This factorization shows that the zeros of the numerator are \(x = 3\) and \(x = -3\). Factorization is an essential step in determining both the zeros and understanding the form of the function itself.

When working with rational functions, another critical concept is the determination of undefined points. This occurs because division by zero is undefined in mathematics.For the function \(g(x) = \frac{x^2 - 9}{x+3}\), the denominator is \(x + 3\).- We set \(x + 3 = 0\) to find which value of \(x\) makes the function undefined. Solving this gives us \(x = -3\). - Since \(x = -3\) makes the denominator zero, the function is undefined at this point. This impacts the identification of zeros. Although \(x = -3\) is a zero of the numerator, it cannot be a zero of the total rational function \(g(x)\) due to the zero denominator at \(x = -3\). Therefore, we exclude \(x = -3\) from the set of zeros for the function. Thus, the only valid zero is \(x = 3\). Understanding undefined points is crucial for correctly determining the behavior and valid zeros of rational functions.