Find the zeros of the polynomial function n(x)=−0.5x^3−1.5x^2+2x, given its graph. Enter the zeros in order from least to greatest.
To find the zeros of the polynomial function, we need to find the values of x for which n(x) equals zero. From the graph, we can see that the polynomial intersects the x-axis at x = -2, x = 0, and x = 3.
Therefore, the zeros of the polynomial function n(x) = -0.5x^3 - 1.5x^2 + 2x are: -2, 0, and 3.
To find the zeros of the polynomial function, we need to set the function equal to zero and solve for x.
n(x) = -0.5x^3 - 1.5x^2 + 2x
Setting n(x) equal to zero:
-0.5x^3 - 1.5x^2 + 2x = 0
Factoring out an x:
x(-0.5x^2 - 1.5x + 2) = 0
Now we have two possible cases:
Case 1: x = 0
Set x equal to zero to get one zero.
x = 0
Case 2: -0.5x^2 - 1.5x + 2 = 0
To find the remaining zeros, we need to solve the quadratic equation -0.5x^2 - 1.5x + 2 = 0.
Using factoring or the quadratic formula, we can solve for x.
Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -0.5, b = -1.5, and c = 2.
x = (-(-1.5) ± √((-1.5)^2 - 4(-0.5)(2))) / (2(-0.5))
Simplifying the equation:
x = (1.5 ± √(2.25 + 4)) / (-1)
x = (1.5 ± √6.25) / (-1)
x = (1.5 ± 2.5) / (-1)
Now we have two possible solutions:
x = (1.5 + 2.5) / (-1) = 4 / (-1) = -4
x = (1.5 - 2.5) / (-1) = -1 / (-1) = 1
Therefore, the zeros of the polynomial function are x = 0, x = -4, and x = 1.
Entering these zeros in order from least to greatest, we have: -4, 0, 1.