To analyze the polynomial \( f(x) = 2x^3(x - 2)^2 \), we can derive its key features, including its roots, multiplicities, end behavior, and any potential turning points.
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Roots and Multiplicities:
- The polynomial can be factored as \( 2x^3(x - 2)^2 \).
- This gives us roots at \( x = 0 \) with multiplicity 3 (odd), and \( x = 2 \) with multiplicity 2 (even).
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End Behavior:
- The leading term is \( 2x^5 \) (since \( 2x^3 \) has 3 and \( (x-2)^2 \) has 2, their degrees add up to 5).
- Since the leading coefficient (2) is positive and the degree of the polynomial is odd, as \( x \to -\infty \), \( f(x) \to -\infty \), and as \( x \to +\infty \), \( f(x) \to +\infty \).
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Behavior at Roots:
- At \( x = 0 \): Since the root has an odd multiplicity (3), the graph will cross the x-axis at this point.
- At \( x = 2 \): Since the root has an even multiplicity (2), the graph will touch (but not cross) the x-axis at this point.
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Summary of Graph Features:
- The graph will start from the bottom left (as \( x \to -\infty \)), cross the x-axis at the origin (0,0), and then rise towards the point (2,0) where it will lower and return to the x-axis but will not cross it (due to the even multiplicity).
- Finally, it will continue to rise towards the top right as \( x \to +\infty \).
With this analysis, a valid sketch of the graph of \( f(x) = 2x^3(x - 2)^2 \) should show:
- An x-intercept at \( x = 0 \) where the graph crosses the axis.
- A touch at \( x = 2 \) where the graph remains on the x-axis without crossing.
- Overall increasing behavior from left to right.
You would need to select a graph from the options provided that demonstrates these characteristics: an x-axis crossing at 0, touching at 2, and the correct end behavior.
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