1. Special Sequences: Polynomial
Dr. Shildneck

2. Polynomial Sequences
A Polynomial Sequence is a pattern of numbers that can be defined by an explicit rule whose formula is a polynomial.

3. Polynomial Sequences To determine if a sequence is polynomial,
Find the difference between terms (1st Difference),
If that is not the same, find the difference between the difference between terms (2nd Difference),
If that is not the same, find the difference between the difference between the difference between terms (3rd Difference),
Continue as needed (4th, 5th, etc. Difference)

4. Polynomial Sequences
If at any point the nth difference becomes constant, the sequence is polynomial in nature.
Furthermore, the degree of the polynomial rule is the same as the level of (common) difference.

5. Determination of Polynomial Sequences
[Example] Determine (a) if the sequence is polynomial in nature.
If so (b) find the degree of the polynomial rule.
-12, -9, -2, 9, 24, 43, 66, …

6. Finding Polynomial Rules
Once you have determine that a sequence is polynomial, to find the explicit rule you can do the following:
Note the nth difference for the degree of the polynomial.
Write a general equation in standard form for the polynomial using the coefficients (a, b, c, d, etc.) as needed.
Substitute consecutive terms (1st, 2nd, 3rd, etc.) for n (x) and an (y) to create a system of equations. (Note: you must have as many equations as you have coefficient-variables.)
Solve the system to determine the coefficients of the equation.

7. Explicit Polynomial Rules
[Example] Find the explicit rule that defines the sequence:
-12, -9, -2, 9, 24, 43, 66, …

8. Finding Polynomial Rules
While you WILL BE REQUIRED TO set up a system of equations to determine the explicit polynomial rule, there is a quicker way of finding the formula.
Note the nth difference for the degree of the polynomial.
Using the S feature in your calculator, set up a table where L1 is the position of the term, and L2 is the term itself.
Using the CALC menu under the S button, choose the appropriate regression (LinReg –Linear/Arithmetic, QuadReg - Quadratic, CubicReg - Cubic, QuartReg - Quartic) based on the degree you determined in Step 1.
Follow the calculation menu with X:L1, and Y:L2.

9. Finding Polynomial Rules
Note:
The Regression menu also allows you store the equation to !
(The names for the equations are under a$F4)
When doing so, you can quickly assess whether your answer is correct by checking the table.
You can also determine terms in other positions by using the `$(value) command.

10. Explicit Polynomial Rules
[Example] Find the explicit rule that defines the sequence:
0, -7, -8, 3, 32, 85, 168, …

11. Assignment
Worksheet 3 – Polynomial Sequences