To analyze the behavior of the graph of the polynomial \( f(x) = 2x^4 - 6x^3 - 8x^2 \) at its zeros, we first need to find the zeros by setting the polynomial equal to zero:
\[ f(x) = 0 \]
First, we can factor out the common term:
\[ f(x) = 2x^2(x^2 - 3x - 4) = 0 \]
Next, we need to solve the quadratic equation \( x^2 - 3x - 4 = 0 \). We can apply the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = -3 \), and \( c = -4 \):
\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-4)}}{2(1)} \]
\[ x = \frac{3 \pm \sqrt{9 + 16}}{2} \]
\[ x = \frac{3 \pm \sqrt{25}}{2} \]
\[ x = \frac{3 \pm 5}{2} \]
This gives us the solutions:
\[ x = \frac{8}{2} = 4 \quad \text{and} \quad x = \frac{-2}{2} = -1 \]
Now, including the factor \( 2x^2 = 0 \), we also find that \( x = 0 \) is a zero of \( f(x) \).
Thus, the zeros of \( f(x) \) are:
- \( x = 0 \)
- \( x = -1 \)
- \( x = 4 \)
Next, we examine the behavior of the graph at each of these zeros:
-
At \( x = 0 \):
- The factor \( 2x^2 \) indicates that this zero has even multiplicity (specifically, multiplicity 2). The graph will touch the x-axis at this point and will not cross it. The graph will go from below the x-axis to above it as it approaches \( x = 0 \).
-
At \( x = -1 \):
- The factor \( x + 1 \) indicates that this zero has odd multiplicity (specifically, multiplicity 1). The graph will cross the x-axis at this point, indicating a change in sign.
-
At \( x = 4 \):
- The factor \( x - 4 \) also indicates that this zero has odd multiplicity (specifically, multiplicity 1). Like at \( x = -1 \), the graph will cross the x-axis at this point, indicating another change in sign.
Summary of Behavior:
- At \( x = 0 \): The graph touches and does not cross the x-axis (local minimum).
- At \( x = -1 \): The graph crosses from above to below the x-axis.
- At \( x = 4 \): The graph crosses from below to above the x-axis.
This analysis gives a visual idea of how the graph behaves near its zeros. To visualize it thoroughly, it's often useful to plot the graph or use graphing software.