To analyze the behavior of the polynomial function \( f(x) = 2x^4 - 6x^3 - 8x^2 \), we need to find its zeros and determine the nature of those zeros (whether the graph touches or crosses the x-axis).
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Finding the zeros: We can start by factoring the polynomial:
\[ f(x) = 2x^2(x^2 - 3x - 4) \]
Now we need to factor \( x^2 - 3x - 4 \):
\[ x^2 - 3x - 4 = (x - 4)(x + 1) \]
Thus, we can rewrite \( f(x) \) as:
\[ f(x) = 2x^2 (x - 4)(x + 1) \]
The zeros of the function are:
- From \( 2x^2 = 0 \) => \( x = 0 \) (with multiplicity 2)
- From \( x - 4 = 0 \) => \( x = 4 \) (with multiplicity 1)
- From \( x + 1 = 0 \) => \( x = -1 \) (with multiplicity 1)
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Determining the behavior at the zeros:
- At \( x = 0 \) (multiplicity 2): The graph touches the x-axis and turns around.
- At \( x = 4 \) (multiplicity 1): The graph crosses the x-axis.
- At \( x = -1 \) (multiplicity 1): The graph crosses the x-axis.
From this analysis, we can conclude:
- The graph touches the x-axis and reverses direction at \( x = 0 \).
- The graph crosses the x-axis at \( x = 4 \) and \( x = -1 \).
Given the multiple-choice options you've provided, the correct option is:
- The graph touches the x-axis and turns around at 0, and crosses the x-axis at 4 and -1.
Feel free to sketch this function on Desmos or any graphing tool to visualize these behaviors accurately!
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