Answer:
To graph the polynomial y = -3(x² - 25)(3x² + 11x + 6), we can start by finding the x-intercepts, also known as zeros. These are the values of x where the polynomial crosses or touches the x-axis.
To find the zeros, we set y equal to zero and solve for x.
So, we have:
-3(x² - 25)(3x² + 11x + 6) = 0
Now, let's factor each quadratic expression:
x² - 25 can be factored as (x - 5)(x + 5)
3x² + 11x + 6 can be factored as (3x + 2)(x + 3)
Now, we substitute these factors back into our equation:
-3(x - 5)(x + 5)(3x + 2)(x + 3) = 0
To find the zeros, we set each factor equal to zero and solve for x:
x - 5 = 0 => x = 5
x + 5 = 0 => x = -5
3x + 2 = 0 => x = -2/3
x + 3 = 0 => x = -3
So, the zeros (x-intercepts) of the polynomial are x = 5, x = -5, x = -2/3, and x = -3.
Now, let's find the midpoints. The midpoint of a line segment is the average of the x-coordinates and the average of the y-coordinates of the endpoints.
Since we only have the x-intercepts, the midpoints would be the average of the x-values only.
The midpoint between 5 and -5 is (5 + (-5))/2 = 0.
The midpoint between -2/3 and -3 is (-2/3 + (-3))/2 = -2.833 (rounded to 3 decimal places).
So, the midpoints are (0, ?) and (-2.833, ?), where ? represents the y-coordinate.
Please note that without the y-values, we can't determine the exact coordinates of the midpoints. However, we have the x-values for the midpoints, which can be helpful in some cases.
Remember to double-check your work and use proper mathematical notation when presenting your solutions.