How To Prove The Division Algorithm For Polynomials Using Induction

Proving 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).

While numerous proofs exist, a rigorous and insightful approach involves mathematical induction on the degree of the dividend polynomial, f(x). This article will guide you through this elegant proof.

Meta Description: Learn how to prove the Division Algorithm for polynomials using mathematical induction. This detailed guide breaks down the process step-by-step, making this complex theorem accessible and understandable.

Base Case: Degree of f(x) = 0

The base case is when the degree of f(x) is 0. This means f(x) is a constant, say f(x) = c, where c is a constant. Let g(x) be a polynomial of degree n ≥ 0.

If n = 0, then g(x) is also a constant, say g(x) = d. We can write c = d(c/d) + 0 if d is not zero. In this case, q(x) = c/d and r(x) = 0. If d is zero, the division is undefined. If n > 0, then we can set q(x) = 0 and r(x) = c. In both cases, the degree of r(x) (0) is less than the degree of g(x). Therefore, the base case holds true.

Inductive Hypothesis: Assume True for Degree k

Let's assume the Division Algorithm holds for all polynomials f(x) with degree k, where k is a non-negative integer. That is, for any such f(x) and g(x) (g(x) ≠ 0), there exist unique polynomials q(x) and r(x) satisfying the conditions stated above.

Inductive Step: Prove True for Degree k+1

Now consider a polynomial f(x) with degree k+1. Let g(x) be a polynomial of degree n, where n ≤ k+1. We can write:

f(x) = a_{k+1}x^{k+1} + a_kx^k + ... + a_1x + a_0, where a_{k+1} ≠ 0.

Let's define f'(x) = f(x) - a_{k+1}x^{k+1} / b_n x^{n-k-1}, where b_n is the leading coefficient of g(x). This reduces the degree of f(x). The degree of f'(x) will be at most k.

By the inductive hypothesis, we can apply the Division Algorithm to f'(x) and g(x):

f'(x) = g(x)q'(x) + r(x), where the degree of r(x) is less than the degree of `g(x)$.

Substituting f'(x) back into the equation:

f(x) - a_{k+1}x^{k+1}/ b_n x^{n-k-1} = g(x)q'(x) + r(x)

Rearranging gives:

f(x) = g(x)q'(x) + r(x) + a_{k+1}x^{k+1}/ b_n x^{n-k-1}

Let q(x) = q'(x) + a_{k+1}x^{k+1}/ b_n x^{n-k-1}. The degree of the term added to q'(x) is carefully chosen. If n ≤ k+1, then the degree of q(x) doesn't increase beyond k+1. The degree of r(x) remains less than the degree of g(x).

Therefore, we have expressed f(x) in the desired form f(x) = g(x)q(x) + r(x), fulfilling the Division Algorithm.

Conclusion

By proving the base case and demonstrating the inductive step, we have successfully shown, using mathematical induction, that the Division Algorithm for polynomials holds true for all polynomials with degree greater than or equal to zero. This fundamental theorem underpins many other significant concepts in abstract algebra and polynomial manipulation. This proof provides a strong foundation for understanding polynomial division and its implications.