To sketch the polynomial \( f(x) = -x(x+2)^2 \), we can analyze its zeros and other characteristics.
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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).
- Solving \( -x(x+2)^2 = 0 \), we find:
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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.
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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 \( x \to -\infty\), \( f(x) \to \infty \).
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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.
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