Question

Find the zeros of the polynomial function f(x) = −0.5x3 −1.5x2+2x given its graph. Enter the zeros in order from least to greatest. (3 points) x= x= x=

Answer 1

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

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

We can factor out a common term, which is \( -0.5x \):

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

Setting each factor equal to zero gives us the first zero:

\[
-0.5x = 0 \implies x = 0
\]

Next, we need to solve the quadratic equation \( x^2 + 3x - 4 = 0 \). We can factor it:

\[
x^2 + 3x - 4 = (x + 4)(x - 1) = 0
\]

Setting each factor to zero:

1. \( x + 4 = 0 \implies x = -4 \)
2. \( x - 1 = 0 \implies x = 1 \)

Now we have three zeros for the polynomial function:

1. \( x = -4 \)
2. \( x = 0 \)
3. \( x = 1 \)

Putting them in order from least to greatest, the zeros are:

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

Thus, the zeros of the polynomial function are:

\[
\text{Zeros: } x = -4, \; x = 0, \; x = 1
\]