To sketch the polynomial \( f(x)=2x^3(x−2)^2 \), we need to analyze its characteristics:
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Degree: The degree of the polynomial is \( 3 + 2 = 5 \). This means the graph will have at most 4 turning points and will approach \( +\infty \) as \( x \to +\infty \) and \( -\infty \) as \( x \to -\infty \) (since the leading coefficient is positive).
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Roots: The roots of the polynomial are found from the factors:
- \( x^3 = 0 \) gives a root at \( x = 0 \) with a multiplicity of 3 (an odd multiplicity means the graph will cross the x-axis at this point).
- \( (x-2)^2 = 0 \) gives a root at \( x = 2 \) with a multiplicity of 2 (an even multiplicity means the graph will only touch the x-axis at this point).
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Behavior at the roots:
- Near \( x = 0 \), the graph will cross the x-axis and head upwards as you move away from the root because of the odd multiplicity.
- Near \( x = 2 \), the graph will touch the x-axis and turn around since the root at \( x = 2 \) has an even multiplicity.
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End behavior:
- As \( x \to -\infty \), \( f(x) \to -\infty \).
- As \( x \to +\infty \), \( f(x) \to +\infty \).
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Sketching the graph:
- Start from the bottom left, crossing at \( x = 0 \) and increasing.
- Approach \( x = 2 \), where the graph touches but does not cross the x-axis, then turns back down before rising again.
With these characteristics in mind, look for a graph that:
- Crosses the x-axis at \( (0, 0) \).
- Touches the x-axis at \( (2, 0) \) without crossing it.
- Goes down towards \( -\infty \) as \( x \) moves left and up towards \( +\infty \) as \( x \) moves right.
If you have images of potential sketches, choose the one that reflects these properties! If you're looking for specific graphs, please describe them, and I can help determine which one matches the polynomial's features.
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