How To Prove The Division Algorithm For Polynomials

How to Prove the Division Algorithm for Polynomials

The division algorithm for polynomials is a fundamental concept in abstract algebra and has wide-ranging applications in various fields, including computer science, engineering, and cryptography. Understanding and proving this algorithm is crucial for anyone working with polynomials. This comprehensive guide will walk you through a detailed proof, exploring its implications and providing practical examples.

Understanding the Division Algorithm

Before diving into the proof, let's clearly state the division algorithm for polynomials. Given two polynomials, f(x) (the dividend) and g(x) (the divisor), where g(x) is not the zero polynomial, there exist unique polynomials q(x) (the quotient) and r(x) (the remainder) such that:

f(x) = q(x)g(x) + r(x)

where the degree of r(x) is less than the degree of g(x), or r(x) is the zero polynomial. This essentially states that any polynomial can be divided by another (non-zero) polynomial, resulting in a quotient and a remainder. The remainder's degree is always smaller than the divisor's degree.

Proving the Division Algorithm: A Step-by-Step Approach

The proof relies on the principle of mathematical induction, a powerful technique for proving statements about integers. We'll adapt it to work with the degrees of polynomials.

Base Case: Degree of f(x) < Degree of g(x)

If the degree of f(x) is less than the degree of g(x), then the division is trivial. We can simply set q(x) = 0 and r(x) = f(x). The equation f(x) = q(x)g(x) + r(x) holds true, and the degree of r(x) (which is equal to the degree of f(x)) is indeed less than the degree of g(x). This establishes our base case.

Inductive Hypothesis: Assumption for Degree n

Now, we assume that the division algorithm holds true for all polynomials f(x) with degree n. This is our inductive hypothesis. In other words, we assume that for any polynomial f(x) of degree n, we can find q(x) and r(x) satisfying the equation and the degree condition.

Inductive Step: Proving for Degree n+1

Let's consider a polynomial f(x) of degree n+1. Let's represent f(x) as:

f(x) = a_n+1xⁿ⁺¹ + a_nxⁿ + ... + a₁x + a₀

where a_n+1 ≠ 0. Let g(x) be our divisor, with degree m. Let's assume the leading term of g(x) is b_mx^m, where b_m ≠ 0.

We can construct a new polynomial:

h(x) = f(x) - (a_n+1/b_m)x^n+1-mg(x)

Observe that the leading term of f(x), a_n+1xⁿ⁺¹, is cancelled out by the leading term of (a_n+1/b_m)x^n+1-mg(x). Therefore, the degree of h(x) is at most n.

Now, because the degree of h(x) is at most n, we can apply our inductive hypothesis. This means there exist polynomials q'(x) and r(x) such that:

h(x) = q'(x)g(x) + r(x)

where the degree of r(x) is less than the degree of g(x).

Substituting the definition of h(x), we get:

f(x) - (a_n+1/b_m)x^n+1-mg(x) = q'(x)g(x) + r(x)

Rearranging the equation, we arrive at:

f(x) = [q'(x) + (a_n+1/b_m)x^n+1-m]g(x) + r(x)

Let's define q(x) = q'(x) + (a_n+1/b_m)x^n+1-m. Then we have:

f(x) = q(x)g(x) + r(x)

This equation satisfies the division algorithm. The degree of r(x) is less than the degree of g(x), as required by our inductive hypothesis.

Conclusion of the Proof

We have successfully shown that if the division algorithm holds for polynomials of degree n, it also holds for polynomials of degree n+1. Combined with the base case (degree less than the divisor's degree), the principle of mathematical induction guarantees that the division algorithm holds for all polynomials with non-negative integer degrees.

Uniqueness of the Quotient and Remainder

The division algorithm not only guarantees the existence of q(x) and r(x) but also their uniqueness. Suppose there are two sets of quotient and remainder, {q₁(x), r₁(x)} and {q₂(x), r₂(x)}, satisfying the algorithm. Then:

f(x) = q₁(x)g(x) + r₁(x) f(x) = q₂(x)g(x) + r₂(x)

Subtracting these equations, we get:

0 = [q₁(x) - q₂(x)]g(x) + [r₁(x) - r₂(x)]

If q₁(x) ≠ q₂(x), then the degree of [q₁(x) - q₂(x)]g(x) is at least the degree of g(x). However, the degree of [r₁(x) - r₂(x)] is less than the degree of g(x). This leads to a contradiction, unless both q₁(x) - q₂(x) and r₁(x) - r₂(x) are zero. This proves the uniqueness of the quotient and remainder.

Practical Applications and Examples

The division algorithm for polynomials is not merely a theoretical concept. It has several practical applications:

1. Finding Roots of Polynomials

The remainder theorem, a direct consequence of the division algorithm, states that when a polynomial f(x) is divided by (x-c), the remainder is f(c). If f(c) = 0, then (x-c) is a factor of f(x), and c is a root. This is extensively used in root-finding algorithms.

2. Partial Fraction Decomposition

In calculus, partial fraction decomposition simplifies complex rational functions into simpler ones, making integration easier. The division algorithm is crucial in ensuring that the degree of the numerator is less than the degree of the denominator before decomposition.

3. Polynomial Interpolation

Polynomial interpolation involves finding a polynomial that passes through a given set of points. Techniques like Lagrange interpolation and Newton's divided difference interpolation rely on polynomial division to construct the interpolating polynomial.

4. Computer Algebra Systems

Modern computer algebra systems heavily rely on polynomial division algorithms for various operations such as simplification, factorization, and solving equations.

Example

Let's divide f(x) = x³ + 2x² - 5x - 6 by g(x) = x - 2.

Using polynomial long division (or synthetic division), we get:

q(x) = x² + 4x + 3 r(x) = 0

Therefore, x³ + 2x² - 5x - 6 = (x - 2)(x² + 4x + 3) + 0. The remainder is 0, indicating that (x-2) is a factor of f(x).

Conclusion

The division algorithm for polynomials is a cornerstone of algebra, offering a structured method for dividing polynomials and providing crucial insights into their properties. Its proof, based on mathematical induction, demonstrates the elegance and power of this fundamental theorem. Understanding this algorithm is essential for anyone working with polynomials, and its practical applications extend far beyond theoretical mathematics, making it a key concept in various scientific and engineering disciplines. The uniqueness of the quotient and remainder further solidifies its importance and applicability in diverse computational contexts. Mastering this concept opens doors to deeper explorations in algebra and its far-reaching applications.