To construct a polynomial expression for a sequence with constant 4th differences of 48, we’ll follow these steps in a way that a 10th grader can understand:
Step 1: Understand the Difference Concept
A sequence is a list of numbers, like: \[ a_0, a_1, a_2, a_3, \ldots \]
The first difference of the sequence is the difference between consecutive terms: \[ a_1 - a_0,, a_2 - a_1,, a_3 - a_2, \ldots \]
The second difference is the difference of the first differences, the third difference is the difference of the second differences, and the fourth difference is the difference of the third differences.
Step 2: Recognize the Degree of Polynomial
When the \(n^{th}\) differences of a sequence are constant, the sequence can be represented by a polynomial of degree \(n\). Since we have constant 4th differences, our polynomial will be of degree 4.
Step 3: Understand the General Form
A general polynomial of degree 4 can be written as: \[ P(x) = ax^4 + bx^3 + cx^2 + dx + e \]
Where \(a\), \(b\), \(c\), \(d\), and \(e\) are constants that we need to find.
Step 4: Use the Constant Difference Information
The constant 4th difference being 48 tells us something specific about the leading coefficient of the polynomial. The formula for the 4th difference (constant) is given by: \[ 24a = 48 \]
Step 5: Solve for \(a\)
From the equation above: \[ 24a = 48 \] To find \(a\): \[ a = \frac{48}{24} = 2 \]
Step 6: Cite the Polynomial
Now that we know \(a\), we can express part of our polynomial: \[ P(x) = 2x^4 + bx^3 + cx^2 + dx + e \] We still need to find the values of \(b\), \(c\), \(d\), and \(e\).
Step 7: Find Additional Terms (if needed)
If you have specific values from the sequence, you can substitute them into the polynomial \(P(x)\) for several values (e.g., \(P(0)\), \(P(1)\), \(P(2)\), ...) to create a system of equations. This will help you solve for \(b\), \(c\), \(d\), and \(e\).
For example:
- \(P(0) = e\)
- \(P(1) = 2(1)^4 + b(1)^3 + c(1)^2 + d(1) + e\)
- \(P(2) = 2(2)^4 + b(2)^3 + c(2)^2 + d(2) + e\)
And so forth.
Step 8: Solve the System of Equations
You should now have a system of equations based on the values of \(P(x)\) for different \(x\). Solving these equations will give you the values of \(b\), \(c\), \(d\), and \(e\).
Conclusion
After you find \(b\), \(c\), \(d\), and \(e\), you’ll have a full polynomial expression that approximates the sequence you started with.
In summary, by knowing the constant 4th difference is 48, we determined part of the polynomial and set the stage to find the remaining coefficients.
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