Answer:
x = -4, 0, and 1.
Explanation:
The zeros of a polynomial function are the values of x where the function equals zero. To find the zeros of the given polynomial function n(x) = -0.5x^3 - 1.5x^2 + 2x, we need to set n(x) equal to zero and solve for x.
-0.5x^3 - 1.5x^2 + 2x = 0
To simplify the equation, we can factor out an x from each term:
x(-0.5x^2 - 1.5x + 2) = 0
Now, we have two possible scenarios for the zeros:
1. x = 0:
When x = 0, the first term becomes zero, so it satisfies the equation.
2. -0.5x^2 - 1.5x + 2 = 0:
To find the zeros for this quadratic equation, we can use factoring, completing the square, or the quadratic formula.
Factoring:
We can try factoring the quadratic expression. However, in this case, it doesn't factor nicely, so we can move on to another method.
Completing the square:
To complete the square, we need to make the coefficient of x^2 equal to 1. We can do this by dividing the entire equation by -0.5.
x^2 + 3x - 4 = 0
Now, we can complete the square. Take half of the coefficient of x (which is 3), square it (which is 9), and add it to both sides of the equation:
x^2 + 3x + 9/4 - 4 - 9/4 = 0
Simplifying, we get:
(x + 3/2)^2 - 25/4 = 0
Next, we can rewrite the equation:
(x + 3/2)^2 = 25/4
Taking the square root of both sides:
x + 3/2 = ±√(25/4)
x + 3/2 = ±(5/2)
Solving for x, we have:
x = -3/2 + 5/2 = 2/2 = 1
x = -3/2 - 5/2 = -8/2 = -4
Therefore, the zeros of the polynomial function n(x) = -0.5x^3 - 1.5x^2 + 2x, in order from least to greatest, are x = -4, 0, and 1.