Question

Use the image to answer the question.

A curve with 2 turning points is plotted on a coordinate grid and is labeled n left parenthesis x right parenthesis. The plot starts in quadrant 2, drops to a turning point in quadrant 3, rises to a turning point in quadrant 1 and drops through quadrant 4.

Find the zeros of the polynomial function n(x)=−0.5x3−1.5x2+2x
, given its graph. Enter the zeros in order from least to greatest.

(2 points)
The zeros of n(x)
are x=
, x=
, and x=
.

Answer 1

To find the zeros of the polynomial function \( n(x) = -0.5x^3 - 1.5x^2 + 2x \), we need to set the function equal to zero and solve for \( x \):

\[
-0.5x^3 - 1.5x^2 + 2x = 0
\]

We can factor out \( -0.5x \):

\[
-0.5x(x^2 + 3x - 4) = 0
\]

Setting each factor equal to zero gives us:

1. \( -0.5x = 0 \)
\[
x = 0
\]

2. For the quadratic \( x^2 + 3x - 4 = 0 \), we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = 3 \), and \( c = -4 \).

Calculating the discriminant:
\[
b^2 - 4ac = 3^2 - 4(1)(-4) = 9 + 16 = 25
\]

Now substituting into the formula:
\[
x = \frac{-3 \pm \sqrt{25}}{2 \cdot 1} = \frac{-3 \pm 5}{2}
\]

This gives us two more roots:
\[
x = \frac{2}{2} = 1 \quad (\text{using the } + \text{ sign}),
\]
\[
x = \frac{-8}{2} = -4 \quad (\text{using the } - \text{ sign}).
\]

Thus, the zeros of \( n(x) \) are \( x = -4 \), \( x = 0 \), and \( x = 1 \).

In order from least to greatest, the zeros are:

\[
\text{The zeros of } n(x) \text{ are } x = -4, x = 0, \text{ and } x = 1.
\]