Let's break down and analyze the graph step by step to address your questions:

(i) Identify the turning points, zeros, and x-intercepts.

• Turning points: Points where the graph changes direction (from increasing to decreasing or vice versa). These are local maxima or minima.

  • Point A: $(-1.569, -3.124)$ (local minimum)
  • Point B: $(0.319, 8.643)$ (local maximum)
  • Point E: $(-1, 0)$ (this appears to be an inflection point, but it coincides with the x-intercept)

• Zeros: Values of $x$ where $f(x) = 0$, i.e., where the graph crosses or touches the x-axis. These are also x-intercepts.

  • $x = -2$ at point D: $(-2, 0)$
  • $x = -1$ at point E: $(-1, 0)$
  • $x = 2$ at point C: $(2, 0)$

  Therefore, the x-intercepts are at $x = -2$, $x = -1$, and $x = 2$.

(ii) Is there any zero with multiplicity?

Yes, the zero at $x = -1$ (point E) appears to have a multiplicity of 2. The reason is that at this point, the graph touches the x-axis and turns around, rather than crossing it, which indicates that the corresponding factor in the polynomial is squared (or has an even power). This is typical behavior for roots with even multiplicity.

(iii) Degree of the polynomial and increasing/decreasing behavior.

• The graph appears to have 3 turning points and crosses the x-axis in 3 distinct places, suggesting that the polynomial is of degree 4. A degree-4 polynomial typically has up to 3 turning points and 4 roots.

• Increasing and decreasing intervals:

  • The polynomial is decreasing in the intervals:
    • From $(-\infty, -1.569)$ (left of point A)
    • From $(0.319, 2)$ (between points B and C)
  • The polynomial is increasing in the intervals:
    • From $(-1.569, 0.319)$ (between points A and B)
    • From $(2, \infty)$ (right of point C)

(iv) Local maximum and minimum:

• Local maximum:

  • At point B: $(0.319, 8.643)$

• Local minimum:

  • At point A: $(-1.569, -3.124)$

(v) Find the remainder when the polynomial is divided by $x - 4$.

By the Remainder Theorem, the remainder when a polynomial $f(x)$ is divided by $x - a$ is equal to $f(a)$. In this case, we need to find $f(4)$, which is the value of the polynomial at $x = 4$.

Since the graph does not directly provide the value of the polynomial at $x = 4$, we would need either the polynomial's equation or more points to estimate $f(4)$. However, based on the visual trends of the graph, $f(4)$ seems to be large and positive, but without the exact equation, we can't provide the precise value of the remainder.

Would you like to go deeper into any part of this explanation or have any additional questions? Here's a list of related questions:

1. Can you determine the exact polynomial equation given the graph?
2. How do you calculate the derivative of the polynomial to find increasing/decreasing intervals?
3. How would the behavior change if the polynomial degree were higher?
4. What is the role of multiplicity in determining the shape of the graph near a zero?
5. How can we compute the remainder using synthetic division?

Tip: When analyzing a polynomial's graph, always check how the graph behaves around the x-axis to identify zero multiplicity (touching vs. crossing).