The Correct Answer and Explanation is:
(x+1)^2(x-4)
To determine the expression for the polynomial, one must analyze its graph to identify its roots and their multiplicities. Since the graph is not provided, this answer is based on a representative example of such a problem. The process described here will allow you to find the correct answer using your specific graph.
The foundation for this problem is the relationship between the roots of a polynomial and its factors. If a polynomial has a root at x = c, then (x – c) is a factor of that polynomial. The roots are the x-values where the graph of the function crosses or touches the x-axis.
Next, you must consider the multiplicity of each root. The multiplicity is indicated by the behavior of the graph at the x-intercept. If the graph crosses straight through the x-axis at a root, the multiplicity is odd, typically 1. This means the corresponding factor, (x – c), will have a power of 1. If the graph touches the x-axis at a root and then turns around (like a parabola at its vertex), the multiplicity is even, typically 2. This means the corresponding factor will have a power of 2, written as (x – c)^2.
For the example answer, (x+1)^2(x-4), we assume a graph with the following features:
- The graph has a root at x = 4. At this point, the graph crosses the x-axis, indicating a multiplicity of 1. This gives us the factor (x – 4).
- The graph has another root at x = -1. At this point, the graph touches the x-axis and turns around, indicating a multiplicity of 2. This gives us the factor (x – (-1))^2, which simplifies to (x + 1)^2.
The final polynomial expression (without the leading coefficient ‘a’) is the product of these factors. Therefore, the expression to fill in the blank is (x+1)^2(x-4). You should apply this same method of identifying roots and their multiplicities from your specific graph.
