Explain the steps to construct an approximate polynomial expression to represent a sequence with constant 4th differences of 48 in 4 simple steps.

Answer 1

Step 1: Determine the degree of the polynomial expression.
To construct an approximate polynomial expression with constant 4th differences, we need to determine the degree of the polynomial. Since the constant 4th differences indicate a degree of polynomial of 4, we will be constructing a quartic polynomial.

Step 2: Find the general expression for a quartic polynomial.
A general quartic polynomial expression can be written as:
P(x) = ax^4 + bx^3 + cx^2 + dx + e

Step 3: Substitute values for x and the corresponding terms of the sequence.
Select four terms from the given sequence and substitute them into the polynomial expression to form a system of equations. For example, if the terms of the sequence are represented as a1, a2, a3, a4, ... etc., then we can choose the terms as a1, a2, a3, and a4.

P(1) = a(1)^4 + b(1)^3 + c(1)^2 + d(1) + e = a + b + c + d + e = a1
P(2) = a(2)^4 + b(2)^3 + c(2)^2 + d(2) + e = 16a + 8b + 4c + 2d + e = a2
P(3) = a(3)^4 + b(3)^3 + c(3)^2 + d(3) + e = 81a + 27b + 9c + 3d + e = a3
P(4) = a(4)^4 + b(4)^3 + c(4)^2 + d(4) + e = 256a + 64b + 16c + 4d + e = a4

Step 4: Solve the system of equations to find the values of the coefficients.
Solve the system of equations obtained in step 3 to find the values of the coefficients a, b, c, d, and e. This can be done by solving the system using algebraic methods such as substitution or elimination.
Once the coefficients are determined, we can write the approximate polynomial expression for the given sequence.