Understanding Differences of Polynomials

When we talk about the differences of a polynomial, we are looking at how the value of the polynomial changes as the input variable \(x\) changes by a constant step size (usually taken as 1 for simplicity, but the principle holds for any constant step size). The first difference is like finding the slope between points on the polynomial's graph, the second difference is the difference of the first differences, and so on.

First Difference of a Polynomial

Let's consider a polynomial of degree \(n\), denoted by \(f(x)\). The first difference, denoted by \(\Delta f(x)\), is defined as:

\(\Delta f(x) = f(x+1) - f(x)\)

If \(f(x)\) is a polynomial of degree \(n\), say \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\), where \(a_n \neq 0\). Then:

\(f(x+1) = a_n (x+1)^n + a_{n-1} (x+1)^{n-1} + \dots + a_1 (x+1) + a_0\)

When we expand \((x+1)^n\), the highest power term is \(x^n\). So, \(f(x+1)\) will have a leading term \(a_n x^n\). When we subtract \(f(x)\), the \(a_n x^n\) term from \(f(x+1)\) (which comes from \(a_n (x+1)^n = a_n (x^n + nx^{n-1} + \dots)\)) will cancel with the \(a_n x^n\) term from \(f(x)\).

The highest power term remaining after subtraction will come from the \(x^{n-1}\) term in the expansion of \((x+1)^n\), which is \(nx^{n-1}\). The leading term in \(\Delta f(x)\) will be \(a_n \cdot nx^{n-1}\).

So, the first difference \(\Delta f(x)\) is a polynomial of degree \(n-1\).

Subsequent Differences

Let's see what happens when we take the second difference, which is the first difference of the first difference:

\(\Delta^2 f(x) = \Delta (\Delta f(x))\)

Since \(\Delta f(x)\) is a polynomial of degree \(n-1\), taking its first difference \(\Delta (\Delta f(x))\) will result in a polynomial of degree \((n-1)-1 = n-2\).

We can generalize this pattern:

• The first difference (\(\Delta f(x)\)) of a polynomial of degree \(n\) has degree \(n-1\).
• The second difference (\(\Delta^2 f(x)\)) of a polynomial of degree \(n\) has degree \(n-2\).
• The third difference (\(\Delta^3 f(x)\)) of a polynomial of degree \(n\) has degree \(n-3\).
• ...
• The \(k\)-th difference (\(\Delta^k f(x)\)) of a polynomial of degree \(n\) has degree \(n-k\).

The nth Difference

Following the pattern, the \(n\)-th difference (\(\Delta^n f(x)\)) of a polynomial of degree \(n\) will have a degree of \(n-n\).

Degree of \(\Delta^n f(x) = n - n = 0\).

A polynomial of degree 0 is a constant.

For example, consider a polynomial of degree 2: \(f(x) = ax^2 + bx + c\), where \(a \neq 0\).

First difference:

\(\Delta f(x) = f(x+1) - f(x)\)

\(\Delta f(x) = [a(x+1)^2 + b(x+1) + c] - [ax^2 + bx + c]\)

\(\Delta f(x) = [a(x^2 + 2x + 1) + bx + b + c] - [ax^2 + bx + c]\)

\(\Delta f(x) = [ax^2 + 2ax + a + bx + b + c] - [ax^2 + bx + c]\)

\(\Delta f(x) = (ax^2 - ax^2) + (2ax + bx - bx) + (a + b + c - c)\)

\(\Delta f(x) = 2ax + a + b\)

This is a polynomial of degree 1.

Second difference (the nth difference in this case since n=2):

\(\Delta^2 f(x) = \Delta (\Delta f(x)) = \Delta (2ax + a + b)\)

\(\Delta^2 f(x) = [2a(x+1) + a + b] - [2ax + a + b]\)

\(\Delta^2 f(x) = [2ax + 2a + a + b] - [2ax + a + b]\)

\(\Delta^2 f(x) = (2ax - 2ax) + (2a + a + b - a - b)\)

\(\Delta^2 f(x) = 2a\)

Since \(a \neq 0\), \(2a\) is a non-zero constant. This is a polynomial of degree 0.

This example illustrates that the nth difference of a polynomial of degree n is a constant.

A polynomial of degree zero is a constant value. Therefore, the nth difference of a polynomial of degree n in x is a constant.

Revision Table: Polynomial Differences

Additional Information: Finite Differences

The concept of differences of polynomials is part of finite differences, which is analogous to differentiation in calculus but applied to discrete values. Instead of looking at the instantaneous rate of change (derivative), finite differences look at the change over a fixed interval.

• Forward Difference: \(\Delta f(x) = f(x+h) - f(x)\)
• Backward Difference: \(\nabla f(x) = f(x) - f(x-h)\)
• Central Difference: \(\delta f(x) = f(x + h/2) - f(x - h/2)\)

For a polynomial of degree \(n\), applying any of these differencing operators \(n\) times will result in a constant value (assuming a constant step size \(h\)). Applying it \(n+1\) times will result in zero.

This property is fundamental in numerical analysis, particularly in areas like interpolation and numerical differentiation and integration.