INVESTIGATION: HOW TO FIND THE LEADING COEFFICIENT USING FINITE DIFFERENCES
In grade 9, you learned how to use the 1st finite differences to determine whether a
function was linear. In grade 10, you learned how to use the 2nd finite differences to
determine whether a function is quadratic. Today, you will learn how to find the degree
of ANY polynomial function using finite differences. By the end of this handout, you
should be able to:
•
Find a relationship between the degree of a polynomial and its finite differences
Part 1: Recalling how to find 1st and 2nd Differences
Example 1: Prove that the given table of values represent linear or quadratic functions
using 1st or 2nd differences.
𝑥
𝑦
0
1
2
3
4
5
9
6
3
0
-3
-6
𝑥
𝑦
0
1
0
9
2
24
3
45
4
72
1st difference
1st difference
2nd difference
Part 2: Do finite differences work for ALL polynomial functions?
Find the 3rd differences for the cubic function 𝑓(𝑥) = 𝑥 3
𝑥
-3
-2
-1
0
1
2
3
𝑦
1st difference
2nd difference
3rd difference
If you are wondering if this applies to ALL polynomial functions, the answer is YES!
The 𝒏th difference are equal for any polynomial function of degree 𝒏.
Test it out yourself! Find the 4th difference for the polynomial function, 𝑓(𝑥) = 𝑥 4 .
Part 3: Relating finite differences to the leading coefficient
Using the 𝑛th finite difference to determine the degree of the polynomial function is nice
but wouldn’t it be nice if we were able to determine the value of the leading
coefficient as well? Well, here you go!
𝑛th difference = 𝑎 × 𝑛!
where
𝑛 represents the degree of the polynomial function
𝑎 represents the leading coefficient
Note: 𝑛! Represents “𝒏 factorial” If 𝒏 = 𝟓, 𝟓! = 𝟓 × 𝟒 × 𝟑 × 𝟐 × 1. If 𝒏 = 𝟑, 𝟑! = 𝟑 × 𝟐 × 𝟏.
Example 2: Find the leading coefficient using the table of values.
𝑥
-3
-2
-1
0
1
2
3
𝑦
-36
-12
-2
0
0
4
18
𝑥
-3
-2
-1
0
1
2
𝑦
7
5
1
-5
-13
-23
Summary: For a polynomial function of degree 𝑛, the 𝑛th differences
•
•
•
Are equal
Have the same sign as the leading coefficient
Are equal to 𝑎 × 𝑛!, where 𝑎 is the leading coefficient
Caution: the formula
𝑛th difference = 𝑎 × 𝑛!
only works when the
𝑥-values in the table of
values is increasing by 1