Explain the steps to construct an approximate polynomial expression to represent a sequence with constant 4th differences of 48 write it like a 10th grader

Answer 1

To construct an approximate polynomial expression for a sequence that has constant fourth differences of 48, follow these steps:

Step 1: Understand the relationship between differences and polynomials

If a sequence has constant \(n\)-th differences, it can be represented by a polynomial of degree \(n\). In this case, since we have constant fourth differences, we are looking for a polynomial of degree 4.

Step 2: Set up the general form of the polynomial

A fourth-degree polynomial can be represented in the general form:

\[ P(n) = an^4 + bn^3 + cn^2 + dn + e \]

Here, \(a\), \(b\), \(c\), \(d\), and \(e\) are constants that we need to determine.

Step 3: Use the constant fourth difference

We know that the fourth difference is constant and equal to 48. The fourth difference of a polynomial \(P(n)\) is given by \(24a\) (the difference formula for polynomials). So we can set up the equation based on what we know:

\[ 24a = 48 \]

Step 4: Solve for the leading coefficient \(a\)

Now, we can solve for \(a\):

\[ a = \frac{48}{24} = 2 \]

Step 5: Set up the polynomial with the found value of \(a\)

Now, we can substitute \(a\) back into the polynomial's general form. Our polynomial now looks like this:

\[ P(n) = 2n^4 + bn^3 + cn^2 + dn + e \]

Step 6: Determine the other coefficients \(b\), \(c\), \(d\), and \(e\)

To find the other coefficients, we typically need some additional information such as specific values of the sequence for certain \(n\). If we have values of the sequence for some \(n\), we can plug those in and create a system of equations to solve for \(b\), \(c\), \(d\), and \(e\).

For example, if we know the values of the sequence for \(n = 0, 1, 2, 3, 4\), we can create equations:

\[ P(0), P(1), P(2), P(3), P(4) \]

These can be plugged into the polynomial and set equal to the known sequence values. Then, you can solve for \(b\), \(c\), \(d\), and \(e\).

Summary

1. Recognize that constant fourth differences mean a fourth-degree polynomial.
2. Determine the leading coefficient \(a\) using the fourth difference (here, \(a = 2\)).
3. Create the polynomial: \(P(n) = 2n^4 + bn^3 + cn^2 + dn + e\).
4. Use known values from the sequence to solve for the remaining coefficients \(b\), \(c\), \(d\), and \(e\).

By following these steps, you can construct an approximate polynomial expression that represents the sequence with constant fourth differences of 48!