Explain the steps to construct an approximate polynomial expression to represent a sequence with constant 4th differences of 48.(4 points)
To construct an approximate polynomial expression to represent a sequence with constant 4th differences of 48, follow these steps:
1. Start by writing down the given sequence.
2. Compute the differences between consecutive terms of the sequence.
3. Repeat step 2 until you obtain a sequence of constant differences.
4. Compute the fourth differences of the sequence obtained in step 3.
5. Since the fourth differences are constant and equal to 48, we can construct a polynomial expression of degree 4 (or higher) that has these fourth differences.
6. Use the method of finite differences to find the coefficients of the polynomial expression. This involves constructing a tabular form with the terms, differences, second differences, third differences, and fourth differences. Fill in the gaps in the tabular form using the known information until you have enough equations to solve for the unknown coefficients.
7. Once you have found the coefficients, construct the polynomial expression by adding the terms with their corresponding coefficients. The polynomial expression should have the form: P(x) = ax^4 + bx^3 + cx^2 + dx + e, where a, b, c, d, and e are the coefficients found in step 6.
Note: The polynomial expression obtained may not perfectly represent the given sequence but it will closely approximate it based on the 4th differences.
To construct an approximate polynomial expression to represent a sequence with constant 4th differences of 48, you can follow these steps:
Step 1: Write down the given sequence.
Write down the sequence for which you want to construct the polynomial expression.
Step 2: Calculate the differences.
Calculate the differences between consecutive terms in the sequence. First, calculate the 1st differences by subtracting each term from its previous term. Then, calculate the 2nd differences by subtracting each 1st difference from its previous 1st difference. Repeat this process for the 3rd and 4th differences.
Step 3: Find a pattern in the differences.
Examine the pattern in the differences. Since the 4th differences are constant at 48, it suggests that the original sequence can be approximated by a 4th degree polynomial.
Step 4: Write the general form of the polynomial expression.
Write the general form of a 4th degree polynomial expression using variables as coefficients:
\[ f(x) = ax^4 + bx^3 + cx^2 + dx + e \]
Step 5: Substitute the values from the sequence.
Substitute the values from the sequence into the polynomial expression to obtain a system of equations. Since we have 5 coefficients (a, b, c, d, and e) in the polynomial expression, we'll need 5 equations to solve for these coefficients. Each equation will correspond to one term from the sequence.
Step 6: Solve the system of equations.
Solve the system of equations to find the values of the coefficients. You can use a variety of methods to solve the system, like substitution, elimination, or matrix methods.
Step 7: Write the final polynomial expression.
Once you have found the values of the coefficients, substitute them into the general form of the polynomial expression from step 4.
For example, if the coefficients are: a = 2, b = 3, c = -1, d = 5, and e = 9, the final polynomial expression would be:
\[ f(x) = 2x^4 + 3x^3 - x^2 + 5x + 9 \]
This expression can be used to approximate the given sequence with constant 4th differences of 48.