C. Polynomial and Rational Functions · Polynomial Functions and Equations

Finite Differences and the Degree of a Polynomial

C1.2About 70 minutes

Learning goals

I can build a finite-difference table from equally spaced data.
I can tell the degree of a polynomial function from the first row of differences that is constant.
I can find the leading coefficient from the constant differences using a · n! (and a · n! · hⁿ when the x-step is h).
I can compare the tables, graphs and equations of linear, quadratic, cubic and quartic functions.

Words to know

finite differences
The differences between consecutive y-values in a table where the x-values are equally spaced.

nth differences
The differences found by subtracting n times: first differences, then differences of those (second differences), and so on.

factorial (n!)
The product n × (n − 1) × ... × 2 × 1. For example, 3! = 6 and 4! = 24.

step (h)
The constant gap between consecutive x-values in a table.

A pattern detector

Remember from Grade 10 and 11: if the x-values in a table go up by equal steps, a linear relation has constant first differences, and a quadratic relation has constant second differences.

That idea keeps going. For a polynomial function of degree n, with equally spaced x-values:

the nth differences are constant (and not zero),
the differences before row n are not constant,
every row after row n is all zeros.

So the first constant row of differences tells you the degree. And the constant value tells you the leading coefficient.

A difference table for y = 2x³ − x. The first and second differences change, but the third differences are all 12, so the function is cubic.

Where does the constant come from?

With a step of 1 in x, the constant nth difference of f(x) = axⁿ + ... is

constant nth difference = a × n!

where n! = n × (n − 1) × ... × 2 × 1. So:

linear: a × 1! = a (the slope, as you know)
quadratic: a × 2! = 2a (you saw this in Grade 10)
cubic: a × 3! = 6a
quartic: a × 4! = 24a

For y = 2x³ − x above, 6a = 6(2) = 12. ✓

If the x-values go up by a step of h instead of 1, the constant becomes a × n! × hⁿ. The lower terms (like −x) never affect the constant row; only the leading term does.

Degree n	Constant row	Constant value (step 1)	Constant value (step h)
1 (linear)	1st differences	a	a · h
2 (quadratic)	2nd differences	2a	2a · h²
3 (cubic)	3rd differences	6a	6a · h³
4 (quartic)	4th differences	24a	24a · h⁴
n	nth differences	a · n!	a · n! · hⁿ

How the degree and leading coefficient show up in a difference table.

Worked example

Example 1: Degree and leading coefficient from a table

Find the degree and leading coefficient of the polynomial function that fits this table.

x = 0, 1, 2, 3, 4, 5 and y = 3, 4, 3, −24, −125, −372.

First differences: 1, −1, −27, −101, −247. Not constant.
Second differences: −2, −26, −74, −146. Not constant.
Third differences: −24, −48, −72. Not constant.
Fourth differences: −24, −24. Constant, so the degree is 4.
The step is 1, so 24a = −24, which gives a = −1.

Degree 4 (quartic), leading coefficient −1. (The table comes from y = −x⁴ + 2x³ + 3.)

Worked example

Example 2: A step that is not 1

Find the degree and leading coefficient.

x = 0, 2, 4, 6, 8 and y = 0, −4, −80, −324, −832.

First differences: −4, −76, −244, −508.
Second differences: −72, −168, −264.
Third differences: −96, −96. Constant, so the degree is 3.
The step is h = 2, so a × 3! × 2³ = −96, that is 48a = −96, and a = −2.

Degree 3, leading coefficient −2. (The data come from y = −2x³ + 3x².) If you forgot the h³ you would get a = −16, which is wrong.

Worked example

Example 3: When the differences never settle

Do the values y = 3, 6, 12, 24, 48, 96 (for x = 0, 1, 2, 3, 4, 5) come from a polynomial function?

First differences: 3, 6, 12, 24, 48.
Second differences: 3, 6, 12, 24. Third differences: 3, 6, 12.
Every row is the previous row again, shifted. Each row doubles, so no row will ever be constant.
Notice that each y-value is twice the one before. That is the signature of an exponential function, y = 3(2)ˣ.

No. No row of differences becomes constant. These data are exponential, not polynomial (you will compare the two families in lesson 4).

Worked example

Example 4: Peeling off the leading term

Find the equation of the cubic function that fits x = 0, 1, 2, 3, 4 and y = 1, 3, 9, 25, 57.

Differences: 2, 6, 16, 32; then 4, 10, 16; then 6, 6. Third differences are 6, so 6a = 6 and a = 1.
Subtract x³ from each y: 1 − 0, 3 − 1, 9 − 8, 25 − 27, 57 − 64 gives 1, 2, 1, −2, −7. What remains is quadratic.
Its differences: 1, −1, −3, −5, then −2, −2, −2. So 2b = −2 and the x² coefficient is −1.
Subtract −x², which means adding x²: 1 + 0, 2 + 1, 1 + 4, −2 + 9, −7 + 16 gives 1, 3, 5, 7, 9. That is linear with slope 2 and y-intercept 1: 2x + 1.
Check x = 3: 27 − 9 + 6 + 1 = 25. ✓

y = x³ − x² + 2x + 1

Typical graphs of polynomial functions of degree 1 to 4. As the degree goes up, the graph can bend more times, and one more row of differences is needed before the values become constant.

Comparing the four families

You can now recognize a polynomial function in three ways:

Numerically: which row of finite differences is constant.
Graphically: a line has no bends, a parabola has one, a cubic can have up to two and a quartic up to three (you will study these turning points next lesson).
Algebraically: the highest power of x in the equation.

Graphing technology (a graphing calculator, Desmos or GeoGebra) is a quick way to see the shapes, but the difference table is a tool you can always use by hand. Real measured data are rarely perfect, so if a row of differences is nearly constant, a polynomial of that degree is a good model.

Just for fun

In the 1820s, Charles Babbage designed a giant mechanical calculator called the Difference Engine. Its whole job was to print tables of polynomial values by adding finite differences, just like you did today, but with thousands of brass gears instead of a pencil.

Hands-on activity

Square pyramids of coins or cubes

You need: about 100 coins, sugar cubes or building bricks of the same size, and paper.

Build a one-layer "pyramid" of 1 cube. Then build a two-layer pyramid: a 2 by 2 square layer with 1 cube on top (5 cubes). Then a three-layer pyramid: 3 by 3, then 2 by 2, then 1 (14 cubes).
Keep going (or count without building) for 4 and 5 layers. Record n (layers) and T (total cubes).
Make a difference table for T. Which row is constant?
Use the constant row to find the degree and leading coefficient of the polynomial T(n).

What to look for: T = 1, 5, 14, 30, 55. Differences: 4, 9, 16, 25; then 5, 7, 9; then 2, 2. Third differences are constant, so T is cubic with 6a = 2, a = 1/3. (The full formula is T(n) = n(n + 1)(2n + 1)/6, whose leading term is (2/6)n³ = (1/3)n³.)

Your turn

x = 0, 1, 2, 3, 4 and y = 5, 8, 11, 14, 17. What type of function fits?

x = −1, 0, 1, 2, 3, 4 and y = −1, 0, 1, 8, 27, 64. Find the degree and leading coefficient.

x = 0, 1, 2, 3, 4, 5 and y = 0, 2, 32, 162, 512, 1250. Find the degree and leading coefficient.

What is the constant third difference of y = −4x³ + 7x − 1 with x-step 1?

The fourth differences of a polynomial function are all −12 (x-step 1). Find the degree and leading coefficient.

Explain why the second differences of a cubic function cannot be constant.

y = 8x³ is tabulated with x = 0, 0.5, 1, 1.5, 2, ... What is the constant third difference?

More practice: warm-up

Which row of differences is constant for a quadratic function?

Which row of differences is constant for a polynomial of degree 5?

What is 4!?

What are the fifth differences of any quartic function?

A linear function has constant first differences of −2 (x-step 1). What is its slope?

More practice: on your own

x = −3, −2, −1, 0, 1, 2, 3 and y = −15, 0, 3, 0, −3, 0, 15. Find the degree and leading coefficient.

The third differences of a cubic are 36, and the x-values go up by 3. Find a.

x = 0, 1, 2, 3, 4 and y = 1, 1, 1, 1, 1. What is the degree?

A ball's height was measured every 0.5 s: 2.0, 4.8, 6.4, 6.7, 5.8, 3.7 m. Are the data close to quadratic?

Do the values y = 1, 4, 9, 16, 25, 36 (x = 1 to 6) and y = 2, 4, 8, 16, 32, 64 (x = 1 to 6) come from polynomials?

Challenge

Show algebraically that the second differences of f(x) = ax² + bx + c (x-step 1) are always 2a.

A cubic function fits x = 0, 1, 2, 3 with y = 2, 1, 4, 17. Find its equation.

Sums of squares 1, 1 + 4, 1 + 4 + 9, ... give 1, 5, 14, 30, 55, 91. Show the formula T(n) = n(n + 1)(2n + 1)/6 has the leading coefficient that the difference table predicts.

Check your understanding

Quiz

For equally spaced x-values, which row of differences is constant for a cubic function?

The fourth differences of a polynomial are constant at 72, with an x-step of 1. What is the leading coefficient?

What is the constant third difference of y = 5x³ − 2x² + x with x-step 1?

A table has x-step 2, and its third differences are constant at 48. What is the leading coefficient of the cubic?

The y-values 2, 6, 18, 54, 162 (x = 0 to 4) come from which kind of function?

A table's second differences are 4, 10, 16, 22 (x-step 1). What can you conclude?

For parents and teachers

Big idea (C1.2): in a table with equally spaced x-values, the degree of a polynomial is the number of the first row of differences that is constant, and the constant equals a × n! (times hⁿ if the step is h).

How to help: Give your teen a table such as y = x³ for x = 0 to 5 (0, 1, 8, 27, 64, 125) and ask them to find the differences out loud. Then ask, "What would change if every y-value were doubled?" (Every difference doubles, so the constant third difference goes from 6 to 12.)

Common difficulties:

Subtracting in the wrong order (always later value minus earlier value), which flips signs.
Using a table whose x-values are not equally spaced; finite differences only work for equal steps.
Forgetting the hⁿ factor when the step is not 1.
Stopping at a row that only looks constant for two entries; check with as many values as you can.