To analyze the behavior of the graph of the polynomial function \( f(x) = 2x^4 - 6x^3 - 8x^2 \), we first need to find its zeros. The zeros of the polynomial can be found by factoring it or using the Rational Root Theorem.
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Finding the Zeros: Start by factoring out the common term: \[ f(x) = 2x^2(x^2 - 3x - 4) \]
Now, we can factor \( x^2 - 3x - 4 \): \[ x^2 - 3x - 4 = (x - 4)(x + 1) \] So, we rewrite \( f(x) \): \[ f(x) = 2x^2(x - 4)(x + 1) \]
This gives us the zeros: \[ x = 0 \quad (multiplicity \ 2), \quad x = 4 \quad (multiplicity \ 1), \quad x = -1 \quad (multiplicity \ 1) \]
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Behavior at the Zeros:
- At \( x = 0 \) (multiplicity 2): The graph touches the x-axis and turns around because the polynomial has an even multiplicity at this zero.
- At \( x = 4 \) (multiplicity 1): The graph crosses the x-axis since the polynomial has an odd multiplicity at this zero.
- At \( x = -1 \) (multiplicity 1): Similar to \( x = 4 \), the graph crosses the x-axis because this zero also has an odd multiplicity.
Summary of behavior at the zeros:
- Touching and turning around at \( x = 0 \)
- Crossing at \( x = 4 \)
- Crossing at \( x = -1 \)
From the responses given in the question, the appropriate choice is:
- The graph crosses the x-axis at 4 and -1, and touches the x-axis and turns around at 0.