Question
Explain the steps to construct an approximate polynomial expression to represent a sequence with constant 4th differences of 48 as a student would(4 points)
Answers
Answered by
GPT-4o mini
To construct an approximate polynomial expression for a sequence that has constant 4th differences of 48, we'll follow these steps:
### Step 1: Identify the Degree of the Polynomial
Since the differences are constant at the 4th level, we know that the sequence can be represented by a polynomial of degree 4. Therefore, we can express the polynomial in the general form:
\[
P(n) = a_4 n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0
\]
### Step 2: Determine the Leading Coefficient
The 4th differences of a polynomial of degree 4 relate to the leading coefficient (a_4) in a specific way. The 4th difference being constant means that the 4th difference of \(P(n)\) is equal to \(4!a_4\). Given that the constant 4th difference is 48, we can set up the equation:
\[
24a_4 = 48
\]
This simplifies to:
\[
a_4 = \frac{48}{24} = 2
\]
### Step 3: Write the Polynomial with Known Coefficient
Now that we have determined \(a_4\), we can write our polynomial as:
\[
P(n) = 2n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0
\]
### Step 4: Determine Remaining Coefficients
Without additional information (such as specific values of the sequence), we cannot uniquely determine the remaining coefficients \(a_3\), \(a_2\), \(a_1\), and \(a_0\). To fully define the polynomial, you would typically need more data points from the sequence to find these coefficients.
For the sake of this exercise, we can leave them as general coefficients \(a_3\), \(a_2\), \(a_1\), and \(a_0\) or assign values according to the particular context or additional constraints given in a problem.
### Final Polynomial Expression
Thus, the polynomial that represents the sequence with constant 4th differences of 48 is:
\[
P(n) = 2n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0
\]
where \(a_3\), \(a_2\), \(a_1\), and \(a_0\) can be found if more information about the sequence is provided.
### Step 1: Identify the Degree of the Polynomial
Since the differences are constant at the 4th level, we know that the sequence can be represented by a polynomial of degree 4. Therefore, we can express the polynomial in the general form:
\[
P(n) = a_4 n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0
\]
### Step 2: Determine the Leading Coefficient
The 4th differences of a polynomial of degree 4 relate to the leading coefficient (a_4) in a specific way. The 4th difference being constant means that the 4th difference of \(P(n)\) is equal to \(4!a_4\). Given that the constant 4th difference is 48, we can set up the equation:
\[
24a_4 = 48
\]
This simplifies to:
\[
a_4 = \frac{48}{24} = 2
\]
### Step 3: Write the Polynomial with Known Coefficient
Now that we have determined \(a_4\), we can write our polynomial as:
\[
P(n) = 2n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0
\]
### Step 4: Determine Remaining Coefficients
Without additional information (such as specific values of the sequence), we cannot uniquely determine the remaining coefficients \(a_3\), \(a_2\), \(a_1\), and \(a_0\). To fully define the polynomial, you would typically need more data points from the sequence to find these coefficients.
For the sake of this exercise, we can leave them as general coefficients \(a_3\), \(a_2\), \(a_1\), and \(a_0\) or assign values according to the particular context or additional constraints given in a problem.
### Final Polynomial Expression
Thus, the polynomial that represents the sequence with constant 4th differences of 48 is:
\[
P(n) = 2n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0
\]
where \(a_3\), \(a_2\), \(a_1\), and \(a_0\) can be found if more information about the sequence is provided.