Explain the steps to construct an approximate polynomial expression to represent a sequence with constant 4th differences of 48 in 4 simple steps.
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.
Step 1: Determine the terms of the sequence
Start by listing the terms of the given sequence. Write down the values of the sequence in order. For example, if the sequence starts with 2, the second term is 4, the third term is 8, and so on, write down these values.
Step 2: Find the first differences
Next, find the first differences of the sequence. Subtract each term from the next term to find the difference. Write these differences in a new column. If the first few first differences are 2, 4, 6, and so on, write down these values.
Step 3: Find the second differences
Continue by finding the second differences. Subtract each first difference from the next first difference to find the second difference. Write these differences in a new column. If the first few second differences are 2, 2, 2, and so on, write down these values.
Step 4: Find the constant fourth difference
Finally, check if the second differences are constant. If the second differences are not constant, continue the process until you find a constant difference. If the constant fourth difference is 48, then you have found your answer. Use this constant difference to construct an approximate polynomial expression to represent the sequence.
Note: The degree of the polynomial will be one less than the number of differences. In this case, since there are four differences, the polynomial will be of degree 3.