Let's analyze the given graph step by step:

(i) Turning Points, Zeros, and X-intercepts

1. Turning Points:

   • $A = (-1.569, -3.124)$
   • $B = (0.319, 8.643)$

2. Zeros/X-intercepts:

   • $D = (-2, 0)$
   • $E = (-1, 0)$
   • $C = (2, 0)$

(ii) Multiplicity

• Multiplicity:
  • At $D = (-2, 0)$, $E = (-1, 0)$, and $C = (2, 0)$, the graph crosses the x-axis, indicating that each zero has an odd multiplicity.
  • Given the polynomial appears to be a cubic function (as observed by the three distinct x-intercepts and two turning points), each zero likely has a multiplicity of 1.

(iii) Degree and Polynomial Behavior

• Degree:

  • The polynomial appears to be cubic (degree 3), due to the presence of two turning points and three x-intercepts.

• Increasing and Decreasing Intervals:

  • Increasing:
    • From $x = -\infty$ to $x = -1.569$
    • From $x = -1$ to $x = 0.319$
    • From $x = 2$ to $x = \infty$
  • Decreasing:
    • From $x = -1.569$ to $x = -1$
    • From $x = 0.319$ to $x = 2$

(iv) Local Maximum and Minimum

• Local Minimum:

  • At $A = (-1.569, -3.124)$

• Local Maximum:

  • At $B = (0.319, 8.643)$

(v) Remainder when Divided by $x-4$

To find the remainder of the polynomial $f(x)$ when divided by $x - 4$, use the Remainder Theorem. The remainder is $f(4)$.

However, since the polynomial function $f(x)$ is not explicitly given in the problem, we cannot directly calculate $f(4)$. If we had the polynomial function, we could substitute $x = 4$ into the polynomial to find the remainder.

Would you like a detailed explanation or any further assistance with these concepts? Here are some questions for you to consider:

1. How do you determine the degree of a polynomial from its graph?
2. What is the significance of the turning points in a polynomial graph?
3. How can you use the Remainder Theorem for polynomial division?
4. What information can you gather from the x-intercepts of a polynomial graph?
5. How can the multiplicity of zeros affect the shape of a polynomial graph?

Tip: When analyzing polynomial graphs, always look for key features such as turning points, intercepts, and the end behavior to understand the degree and nature of the polynomial.