Question

Find the zeros of the polynomial function
grid points (-4,0) (0,0) (2,-5)
n(x)=−0.5x3−1.5x2+2xn\left(x\right)=-0.5x^3-1.5x^2+2xn(x)=−0.5x
3
−1.5x
2
+2x, given its graph.

Enter the zeros (numbers only) in order from least to greatest and separated with a comma.

Answer 1

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

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

We can factor out \( x \) from the equation:

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

This gives us one zero:

\[
x = 0
\]

Now we need to find the zeros of the quadratic \( -0.5x^2 - 1.5x + 2 = 0 \). We'll multiply the entire equation by -2 to make it easier to handle:

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

Next, we can factor this quadratic:

\[
(x + 4)(x - 1) = 0
\]

From this, we find the other two zeros:

\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]

\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]

So, the zeros of the polynomial function are:

\[
x = -4, \quad x = 0, \quad x = 1
\]

In order from least to greatest, the zeros are:

\[
-4, 0, 1
\]

Thus, the final answer is:

\[
-4, 0, 1
\]