How To Prove The Division Algorithm For Polynomials Using Induction

How to Prove the Division Algorithm for Polynomials Using Induction

The Division Algorithm for polynomials is a fundamental theorem in algebra, stating that for any polynomials f(x) and g(x) (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) = g(x)q(x) + r(x)

where the degree of r(x) is less than the degree of g(x). This is analogous to the division algorithm for integers. While the algorithm itself is relatively straightforward to apply, proving it rigorously, particularly using mathematical induction, provides a deeper understanding of its underlying structure. This article will delve into a detailed, step-by-step proof of the Division Algorithm for polynomials using induction on the degree of the dividend polynomial f(x).

Understanding the Basis of the Proof: Mathematical Induction

Before diving into the proof, let's refresh our understanding of mathematical induction. This powerful proof technique involves two crucial steps:

• Base Case: We prove the statement holds true for the smallest possible value (in this case, the lowest degree of f(x)).
• Inductive Step: We assume the statement is true for some arbitrary value k (the inductive hypothesis) and then prove that it must also be true for k+1.

By successfully completing both steps, we establish the truth of the statement for all values greater than or equal to the base case.

Proving the Division Algorithm by Induction: A Step-by-Step Guide

Our proof will proceed by induction on the degree of f(x), denoted as deg(f(x)).

1. Base Case: deg(f(x)) = 0

When deg(f(x)) = 0, f(x) is a constant polynomial, say f(x) = c, where c is a constant. Let g(x) be any non-zero polynomial. We can write:

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

If deg(g(x)) > 0, we can choose q(x) = 0 and r(x) = c. The degree of r(x) (which is 0) is less than the degree of g(x), satisfying the conditions of the Division Algorithm. If deg(g(x)) = 0, then g(x) is also a constant, say g(x) = d. Then we can choose q(x) = c/d and r(x) = 0. This also satisfies the conditions of the division algorithm. Therefore, the base case holds true.

2. Inductive Hypothesis: Assume True for deg(f(x)) = k

Now, we assume that the Division Algorithm holds for all polynomials f(x) with deg(f(x)) = k, where k is a non-negative integer. That is, for any polynomial f(x) of degree k and any non-zero polynomial g(x), there exist unique polynomials q(x) and r(x) such that:

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

with deg(r(x)) < deg(g(x)).

3. Inductive Step: Prove True for deg(f(x)) = k+1

Let f(x) be a polynomial of degree k+1, and let g(x) be a non-zero polynomial. Let the leading term of f(x) be a_k+1x^k+1 and the leading term of g(x) be b_mx^m, where m is the degree of g(x).

We can construct a new polynomial f'(x) as follows:

f'(x) = f(x) - (a_k+1/b_m)x^k+1-mg(x)

Observe that the leading term of (a_k+1/b_m)x^k+1-mg(x) is precisely a_k+1x^k+1, which cancels out the leading term of f(x). Therefore, deg(f'(x)) < k+1, meaning deg(f'(x)) ≤ k.

Now, by our inductive hypothesis, since deg(f'(x)) ≤ k, we can apply the Division Algorithm to f'(x) and g(x):

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

where deg(r(x)) < deg(g(x)).

Substituting the expression for f'(x), we get:

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

Rearranging the equation, we obtain:

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

Let q(x) = q'(x) + (a_k+1/b_m)x^k+1-m. Then we have:

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

with deg(r(x)) < deg(g(x)). This demonstrates that the Division Algorithm holds for polynomials of degree k+1.

4. Conclusion

We have successfully shown that the Division Algorithm for polynomials holds for the base case (deg(f(x)) = 0) and that if it holds for deg(f(x)) = k, it also holds for deg(f(x)) = k+1. By the principle of mathematical induction, we conclude that the Division Algorithm for polynomials is true for all non-negative integer degrees of f(x). This completes the proof.

Further Considerations and Applications

The Division Algorithm for polynomials is a cornerstone of many algebraic concepts and techniques. Here are some of its key applications:

• 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). This allows us to efficiently test for roots of polynomials.

• Polynomial Factorization: The Division Algorithm plays a vital role in factoring polynomials. If we know a factor of a polynomial, we can use division to find the other factors.

• Partial Fraction Decomposition: This technique, crucial in calculus and other areas, relies on the Division Algorithm to express rational functions as sums of simpler fractions.

• Numerical Analysis: The algorithm underlies various numerical methods for approximating roots and solving polynomial equations.

• Abstract Algebra: The concept extends to more abstract settings in abstract algebra, where it provides fundamental tools for working with polynomial rings over various fields.

This comprehensive proof, along with its illustrative examples and applications, should solidify your understanding of the Division Algorithm for polynomials and its significant role in mathematics. Remember that understanding the underlying principles of mathematical proofs is as crucial as applying the algorithms themselves. By grasping the inductive proof, you gain a deeper appreciation of the structural elegance and power of this fundamental theorem.