Answer:
y = 0.96(x -√5)(x + √5).
Explanation:
The two zeros of this polynomial are (0, 0) and (5, 0), and the vertex is (2.5, -6), all of this info readily available from the graph.
The standard equation of a vertical parabola with vertex at (h, k) is
y - k = a(x - h)^2, where the vertex is represented by (h, k). Since we know that the vertex is (2.5, -6), the equation for the given graphed parabola is
y + 6 = a(x - 2.5)^2
Since the graph of this parabola goes through the origin, (0, 0), the following must be true if we substitute 0 for both x and y:
0 + 6 = a(0 - 2.5)^2, or 6 = a(6.25). Thus, a = 6/6.25, or a = 0.96.
The equation of this polynomial must then be:
y + 6 = 0.96(x - 2.5)^2, which is satisfied by both (0, 0) and (5, 0).
This equation can be rewritten as
y = 0.96(x^2 - 5x + 6.25) - 6 = 0.96x^2 - 4.8x + 6 - 6, or just
y = 0.96x^2 - 4.8x. We must factor this. Factoring out 0.96, we get:
y = 0.96(x^2 - 5), which can be factored further:
y = 0.96(x -√5)(x + √5). This is the fully factored form of the polynomial.