To analyze the behavior of the polynomial \( f(x) = 2x^4 - 6x^3 - 8x^2 \) at its zeros, we first need to find the zeros of the polynomial.
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Finding Zeros: We can factor the polynomial \( f(x) \).
\[ f(x) = 2x^2 (x^2 - 3x - 4) \]
Now, we can factor \( x^2 - 3x - 4 \) further: \[ x^2 - 3x - 4 = (x - 4)(x + 1) \] Thus, \[ f(x) = 2x^2 (x - 4)(x + 1) \]
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Identifying the Zeros: Setting \( f(x) = 0 \), we find the zeros:
- 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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Behavior of the Graph at Zeros:
- At \( x = 0 \): Since it has an even multiplicity (2), the graph will touch the x-axis at this point and turn around.
- At \( x = 4 \) and \( x = -1 \): Both have odd multiplicities (1), so the graph will cross the x-axis at these points.
Thus, to summarize the behavior of the polynomial:
- The graph touches the x-axis and turns around at \( x = 0 \).
- The graph crosses the x-axis at \( x = 4 \) and \( x = -1 \).
Given these analyses, the correct response to how the graph behaves at the zeros of the polynomial is:
The graph crosses the x-axis at 4 and -1, and touches the x-axis and turns around at 0.
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