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 (roots) of this polynomial.
Step 1: Factor the polynomial
First, we can factor out the common term:
\[ f(x) = 2x^2(x^2 - 3x - 4) \]
Next, we can factor the quadratic \( x^2 - 3x - 4 \):
\[ x^2 - 3x - 4 = (x - 4)(x + 1) \]
So, the fully factored form of the polynomial is:
\[ f(x) = 2x^2(x - 4)(x + 1) \]
Step 2: Find the zeros
Setting \( f(x) = 0 \), we find the zeros:
- From \( 2x^2 = 0 \) → \( x = 0 \) (with a multiplicity of 2)
- From \( (x - 4) = 0 \) → \( x = 4 \) (with a multiplicity of 1)
- From \( (x + 1) = 0 \) → \( x = -1 \) (with a multiplicity of 1)
So, the zeros are at \( x = 0 \), \( x = 4 \), and \( x = -1 \).
Step 3: Determine the behavior at each zero
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At \( x = 0 \):
- The zero has a multiplicity of \( 2 \) (even). This means the graph will touch the x-axis at \( x = 0 \) and turn around.
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At \( x = 4 \):
- The zero has a multiplicity of \( 1 \) (odd). This means the graph will cross the x-axis at \( x = 4 \).
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At \( x = -1 \):
- The zero has a multiplicity of \( 1 \) (odd). This means the graph will also cross the x-axis at \( x = -1 \).
Conclusion
Based on our analysis, the correct option describing the behavior of the graph at the zeros is:
A. The graph crosses the x-axis at 4 and -1, and touches the x-axis and turns around at 0.