Question

graph f (x) = 2x^4 - 6x^3 - 8x^2. How does the graph behave at zeros of the polynomial

Answer 1

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:
1. \( x = 0 \)
2. \( x = -1 \)
3. \( x = 4 \)

Next, we examine the behavior of the graph at each of these zeros:

1. **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 \( x = 0 \).

2. **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.

3. **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.