To determine which graph corresponds to the polynomial \( f(x) = -x(x+2)^2 \), we can analyze its zeros and behavior.
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Finding the Zeros:
Set \( f(x) = 0 \): \[ -x(x+2)^2 = 0 \] This gives us the zeros:- \( x = 0 \) (from \(-x = 0\))
- \( (x + 2)^2 = 0 \) leads to \( x = -2 \) (with a multiplicity of 2).
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Interpret the Zeros:
- The zero at \( x = 0 \) has a multiplicity of 1, which means the graph will cross the x-axis at this point.
- The zero at \( x = -2 \) has a multiplicity of 2, which means the graph will touch the x-axis at this point but will not cross it.
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Behavior as \( x \) Approaches Positive and Negative Infinity:
Since the leading term (from expanding) will be negative (due to the negative sign in front):- As \( x \to +\infty \), \( f(x) \to -\infty \).
- As \( x \to -\infty \), \( f(x) \to +\infty \).
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Graph Characteristics:
- The graph will start high (positive) in the left quadrant.
- It will touch the x-axis at \( x = -2 \) (turning back up).
- It will cross the x-axis at \( x = 0 \) (going down).
- Finally, it will continue downward into the fourth quadrant.
Using these properties, you can match the description of the graph with those provided. Look for a graph that:
- Touches the x-axis at \( x = -2 \).
- Crosses the x-axis at \( x = 0 \).
- Falls to the right and rises to the left.
This should lead you to identify the correct sketch of the polynomial.
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