To construct a polynomial expression that represents a sequence with constant fourth differences of 48, we can follow these steps:
Step 1: Understand the polynomial degree
The fact that the sequence has constant fourth differences implies that the corresponding polynomial will be of degree 4. This means the polynomial can be expressed 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: Analyze the fourth difference
Given that the constant fourth difference is 48, we can use this information to determine the leading coefficient \(a_4\). The fourth difference of a polynomial of degree \(d\) is given by:
\[ \Delta^4 P(n) = 4! a_4 = 24 a_4 \]
Since we are told that the fourth difference is 48, we can set up the equation:
\[ 24 a_4 = 48 \]
Solving for \(a_4\):
\[ a_4 = \frac{48}{24} = 2 \]
Step 3: Establish the polynomial form
At this point, we know that the polynomial begins as follows:
\[ P(n) = 2n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0 \]
Step 4: Use initial values or conditions
To fully determine the coefficients \(a_3\), \(a_2\), \(a_1\), and \(a_0\), you will need additional information, such as specific values of the sequence. For instance, if you have values for \(P(1)\), \(P(2)\), \(P(3)\), etc., you could set up a system of equations to solve for these coefficients.
Step 5: Solving for remaining coefficients
Assuming you have values for the first few terms of the sequence (let’s denote them as \(b_1, b_2, b_3, b_4, \ldots\)), you would set up the equations:
- \(P(1) = b_1\)
- \(P(2) = b_2\)
- \(P(3) = b_3\)
- \(P(4) = b_4\)
You can substitute \(n = 1, 2, 3, 4\) into the polynomial and form a system of 4 equations in terms of \(a_3\), \(a_2\), \(a_1\), and \(a_0\).
Step 6: Solve the system of equations
Using methods such as substitution, elimination, or matrix operations, solve the system to find \(a_3\), \(a_2\), \(a_1\), and \(a_0\).
Final Polynomial Expression
Once you have all coefficients determined, you can write the complete polynomial expression representing the sequence:
\[ P(n) = 2n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0 \]
Summary
To summarize, the key steps involve determining that the polynomial is degree 4 due to the constant 4th differences, calculating the leading coefficient, and using provided values to find the other coefficients. From there, you can write the polynomial that approximates the sequence with constant fourth differences of 48.