To determine how the graph of the polynomial \( f(x) = 2x^4 - 6x^3 - 8x^2 \) behaves at its zeros, we first need to find the roots of the polynomial.
Step 1: Factoring the polynomial.
We can start by factoring out the common term: \[ f(x) = 2x^2(x^2 - 3x - 4). \]
Next, we can factor the quadratic further: \[ x^2 - 3x - 4 = (x - 4)(x + 1). \]
Thus, we have: \[ f(x) = 2x^2(x - 4)(x + 1). \]
Step 2: Identify the zeros.
The zeros of the polynomial are:
- \( x = 0 \) (with multiplicity 2),
- \( x = 4 \) (with multiplicity 1),
- \( x = -1 \) (with multiplicity 1).
Step 3: Determine the behavior at the zeros:
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At \( x = 0 \) (multiplicity 2):
- Since the multiplicity is even, the graph will touch the x-axis at this point and turn around.
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At \( x = 4 \) (multiplicity 1):
- Since the multiplicity is odd, the graph will cross the x-axis at this point.
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At \( x = -1 \) (multiplicity 1):
- Similarly, since the multiplicity is odd, the graph will also cross the x-axis at this point.
Putting this all together, we can conclude:
- The graph touches the x-axis and turns around at \( x = 0 \).
- The graph crosses the x-axis at \( x = 4 \) and \( x = -1 \).
Based on the analysis, the correct response 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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