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To begin finding the zeros of a rational function, you need to understand what the zero of a function means. The zero of a function is the value of the variable, typically denoted as \(x\), that makes the function's value equal to zero. When dealing with rational functions, where a polynomial is divided by another polynomial, setting the function to zero means setting the numerator equal to zero. This is because a fraction is zero only when its numerator is zero, assuming the denominator is not zero. In our function \(f(x) = \frac{5x+3}{2x+2}\), setting \(f(x)\) to zero gives us the equation \(0 = \frac{5x + 3}{2x + 2}\). The next crucial step is to solve this equation for \(x\).
By setting the numerator, \(5x + 3\), equal to zero, we begin the process of finding the function's zeros, moving into solving the linear equation that results.

Once you have set the numerator of the rational function to zero, you might find yourself needing to solve a linear equation. Linear equations are relatively straightforward, involving variables raised to the first power. In our example, the resulting equation from setting the function to zero is \(0 = 5x + 3\).

To solve for \(x\), start by isolating the terms involving \(x\) on one side and constants on the other. Subtract 3 from both sides, resulting in \(-3 = 5x\). Next, to isolate \(x\), you divide both sides by the coefficient of \(x\), which is 5 in this case. This gives \(x = \frac{-3}{5}\).
It's crucial to move step-by-step through these arithmetic operations to ensure accuracy. Solving linear equations involves basic algebra, primarily addition, subtraction, multiplication, and division, which are essential skills in finding zeros.

The zero of a function is the value that makes the function equal to zero. For the rational function \(f(x) = \frac{5x+3}{2x+2}\), we determined the zero is at \(x = \frac{-3}{5}\). This means the function crosses the x-axis at this point on a graph. Understanding this concept helps in graphing and analyzing the behavior of functions.

In broader terms, finding the zeros can reveal critical points where a function changes signs, potentially indicating where a graph transitions from positive to negative values or vice versa. It's important to note that the denominator \(2x + 2\) in our function never equals zero here, as that would make the function undefined. However, always consider the possibility of undefined points in more complex rational functions.
Knowing how to find and interpret zeros is a foundational aspect of working with functions, laying groundwork for understanding deeper mathematical concepts.