Calculus How To

Polynomial Sequence

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A polynomial sequence can be generated by various degree polynomials. As there are an infinite number of any particular type of polynomial, there are an infinite number of possible polynomial sequences.

Polynomial Sequence Examples

These sequences are usually integer valued (i.e. their inputs are 1, 2, 3, …).
Examples:

Degree Generating Function Polynomial Sequence
1 f(x) = 3x {3, 6, 9,…}
2 f(x) = 2x2 {2, 8, 18, 32, …}
3 f(x) = 4x3 {4, 32, 108, 256, …}
4 f(x) = x4 + 1 {2, 17, 82, 257, …}
5 f(x) = x5 – 99 {-98, -67, 144, 925, …}

Finding a Formula for a Polynomial Sequence

One way to identify the generating polynomial function is to plot points on a graph.

Example question: What function generates the polynomial sequence {0, 1, 4, 7,…}?

Solution:
Step 1: Make a table of x and y values. Your x-values are the places of each term (1, 2, 3, 4) and your y-values are the terms of the sequence: {0, 1, 4, 7,…}.

x y
1 0
2 1
3 4
4 7

Step 2: Sketch a graph of the points from Step 1:
polynomial sequence on a graph

Step 3: Find the formula: From the graph, it’s clear that this sequence is generated by a linear function. The formula for a linear function is y = mx + b.
The slope is the common difference between the points. The common difference is 3, so m = 3.
“b” is the y-intercept. For this graph, it looks like that’s at 1. So our formula is:
f(x) = 3x + 1

Step 4: Test a couple of points in the formula. Plugging in a couple of points to the formula will confirm the formula you found in Step 3 is correct.

  • f(0) = 3(0) + 1 = 1
  • f(1) = 3(1) + 1 = 4.

The function generating the polynomial sequence is f(x) = 3x + 1.


That’s it!

finding the Degree of the Generating Polynomial

Finding the common difference is the key to finding out which degree polynomial generated any particular sequence. In general, keep taking differences until you get a constant in a row. The number of times you have to take differences is the degree of your polynomial.
Example question: What is the degree of the polynomial that generated the sequence {2, 8, 18, 32}?


Solution: Find the differences between terms:
6 10 14
And again:
6 6.
We had to find the common difference twice to get a constant row, so the polynomial sequence {2, 8, 18, 32} was generated by a second-degree polynomial.

References

Graph: Desmos.com.

CITE THIS AS:
Stephanie Glen. "Polynomial Sequence" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/polynomial-sequence/
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