https://file.aiquickdraw.com/mathbot/file/172546351858648xj069i.jpg

Let's analyze the image and extract the problems step-by-step for solving:

### Extracted Problems:

1. **Problem 1:**
   - Create the following table of values where `y1`, `y2`, `y3`, and `y4` are the first, second, third, and fourth differences of the function.
   - The given table seems to have the following values:

   | x  | f(x) |
   |----|------|
   | 1  | 10   |
   | 2  | 21   |
   | 3  | 36   |
   | 4  | 55   |
   | 5  | 78   |

2. **Problem 2:**
   - Find a cubic polynomial which fits the following table:

   | x  | y  |
   |----|----|
   | 1  | 3  |
   | 2  | 6  |
   | 3  | 19 |
   | 4  | 58 |

3. **Problem 3:**
   - Determine if a polynomial fits the data and find the values if it does:

   | x  | y  |
   |----|----|
   | 0  | 6  |
   | 1  | 11 |
   | 2  | 18 |
   | 3  | 27 |
   | 4  | 38 |

Let's solve these problems step-by-step:

### Solution 1: Difference Table

To create the difference table, we calculate the first, second, third, etc., differences between consecutive values of `f(x)`.

| x  | f(x) | y1  | y2  | y3  | y4  |
|----|------|------|------|------|------|
| 1  | 10   | 11   | 4    | 2    | 0    |
| 2  | 21   | 15   | 6    | 0    |      |
| 3  | 36   | 19   | 6    |      |      |
| 4  | 55   | 23   |      |      |      |
| 5  | 78   |      |      |      |      |

### Solution 2: Cubic Polynomial Fitting

To find a cubic polynomial of the form:

\[
y = ax^3 + bx^2 + cx + d
\]

We set up the system of equations using the provided values and solve for \(a, b, c,\) and \(d\).

### Solution 3: Quadratic Polynomial Fitting

For the given table, we try to fit it using a quadratic polynomial of the form:

\[
y = ax^2 + bx + c
\]

By setting up a system of linear equations using the given points, we can determine if a quadratic polynomial fits the data and solve for the coefficients \(a, b,\) and \(c\).

---

Let's proceed with detailed calculations for each of these solutions.

#### Solution 1: Calculating Differences (Detailed)
Let's compute the values for the first problem to fill out the differences:

1. **First Differences (y1):**
   - \(y1_2 = f(2) - f(1) = 21 - 10 = 11\)
   - \(y1_3 = f(3) - f(2) = 36 - 21 = 15\)
   - \(y1_4 = f(4) - f(3) = 55 - 36 = 19\)
   - \(y1_5 = f(5) - f(4) = 78 - 55 = 23\)

2. **Second Differences (y2):**
   - \(y2_3 = y1_3 - y1_2 = 15 - 11 = 4\)
   - \(y2_4 = y1_4 - y1_3 = 19 - 15 = 4\)
   - \(y2_5 = y1_5 - y1_4 = 23 - 19 = 4\)

3. **Third Differences (y3):**
   - \(y3_4 = y2_4 - y2_3 = 4 - 4 = 0\)
   - \(y3_5 = y2_5 - y2_4 = 4 - 4 = 0\)

4. **Fourth Differences (y4):**
   - \(y4_5 = y3_5 - y3_4 = 0 - 0 = 0\)

Would you like me to provide detailed steps for solving the other two problems or focus on a specific part? 

### Relative Questions
1. How do we find the equation of a polynomial that fits a set of data points?
2. What is the significance of difference tables in polynomial interpolation?
3. How can we solve a system of linear equations to find polynomial coefficients?
4. What are some real-world applications of polynomial interpolation?
5. How does Newton's Divided Difference method differ from Lagrange Interpolation?

### Tip:
When working with polynomials and fitting curves to data, always double-check the degree of the polynomial to ensure it appropriately matches the number of data points and the observed behavior of the data.

Learn how to construct difference tables and find polynomials that fit given data points. This problem covers essential algebraic concepts such as Newton's Divided Difference method and polynomial interpolation, suitable for Grades 9-12 students.

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