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To find the x-intercepts of a polynomial function, we need to determine the values of the variable that make the function equal to zero. In simpler terms, these are the points where the graph of the function crosses the x-axis. For a given polynomial function like \[ g(n) = -2(3n - 1)(2n + 1) \], the x-intercepts are found by setting the function equal to zero and solving for "n". Here's how you do it:1. Set the function to zero: \( g(n) = 0 \).2. Break it down by setting each factor equal to zero: - First factor: \( 3n - 1 = 0 \) - Second factor: \( 2n + 1 = 0 \).3. Solve each equation: - For \( 3n - 1 = 0 \), solve to get \( n = \frac{1}{3} \). - For \( 2n + 1 = 0 \), solve to get \( n = -\frac{1}{2} \).Therefore, the x-intercepts of the function \( g(n) \) are at \( n = \frac{1}{3} \) and \( n = -\frac{1}{2} \). These are the points where the curve touches the x-axis.

The y-intercept of a function provides a way to understand where a function crosses the y-axis. To find this point, simply set the variable equal to zero and solve for the function's value. This is because the y-axis is defined by the line where the variable (in our case "n") is zero.For the function\[ g(n) = -2(3n - 1)(2n + 1) \],here's how you find the y-intercept:1. Substitute \( n = 0 \) into the function: \[ g(0) = -2(3 \cdot 0 - 1)(2 \cdot 0 + 1) \]. 2. Simplify the expression: - \( 3 \cdot 0 - 1 = -1 \) - \( 2 \cdot 0 + 1 = 1 \) - So, \( g(0) = -2(-1)(1) = 2 \).Thus, the y-intercept of the function is at \( g(0) = 2 \). This shows us that the graph crosses the y-axis at the point (0, 2). Seeing where a function intersects the y-axis is important for fully understanding its behavior.

The factored form of a polynomial function tells us a lot about the function's roots, or solutions. It is an expression written as a product of its factors, often helping to easily find intercepts and zeros of the function.Let's consider the function\[ g(n) = -2(3n - 1)(2n + 1) \].Here's how this form is helpful:- **Identifies Roots Quickly**: Each factor of the function corresponds to a potential root by setting it equal to zero. - For example, factor \( 3n - 1 \) suggests a root at \( n = \frac{1}{3} \). - Factor \( 2n + 1 \) suggests a root at \( n = -\frac{1}{2} \).- **Describes the Polynomial's Behavior**: - The leading coefficient (here, \(-2\)) affects the graph's direction. - Factors guide understanding of how the graph crosses or touches axes.Writing a polynomial in its factored form enhances comprehension by breaking down complex expressions into simpler parts. This makes it easier to apply mathematical strategies for sketching graphs and analyzing polynomial behavior.