Question

Use the zeros to determine which of the following graphs is a sketch of the polynomial f(x)=−x(x+2)2 .(1 point)

Answer 1

To sketch the polynomial \( f(x) = -x(x+2)^2 \), we can analyze its zeros and other characteristics.

1. **Zeros/Roots**: The zeros of the polynomial are the values of \( x \) for which \( f(x) = 0 \).
- Solving \( -x(x+2)^2 = 0 \), we find:
- \( -x = 0 \) gives \( x = 0 \).
- \( (x+2)^2 = 0 \) gives \( x+2 = 0 \) or \( x = -2 \) (with a multiplicity of 2).

Therefore, the zeros are \( x = 0 \) (multiplicity 1) and \( x = -2 \) (multiplicity 2).

2. **Multiplicity Effect**:
- A root with odd multiplicity (like \( x = 0 \)) will cross the x-axis.
- A root with even multiplicity (like \( x = -2 \)) will touch the x-axis but not cross it.

3. **Leading Coefficient**: The leading coefficient of the polynomial is negative (because of the negative sign in front). Hence, as \( x \to \infty\), \( f(x) \to -\infty \), and as \( x \to -\infty\), \( f(x) \to \infty \).

4. **Behavior at the Zeros**:
- At \( x = 0 \), the graph will cross the x-axis.
- At \( x = -2 \), the graph will touch the x-axis and turn back.

Now combining all of this:

- The graph should start from \(\infty\) as \( x \to -\infty\).
- The graph will touch at \( x = -2 \) and return.
- Then it will cross the x-axis at \( x = 0\) and go down towards \(-\infty\) as \( x \to \infty\).

Based on this analysis, you can now identify the correct sketch of the polynomial \( f(x) = -x(x+2)^2 \) among the options you have. The correct graph should exhibit these properties.