Question

Whole numbers are closed under addition because the sum of two whole numbers is always a whole number. Explain how the process of checking polynomial division supports the fact that polynomials are closed under multiplication and addition.

Answer 1

When you perform polynomial long division (or synthetic division), every step of the algorithm uses only multiplication and addition/subtraction of polynomials. That is exactly why the algorithm both works and shows closure under those operations.

Sketch of the division algorithm and why it stays inside the set of polynomials:

- Start with two polynomials f(x) (the dividend) and g(x) (the divisor, g ≠ 0). You form a quotient q(x) and remainder r(x) so that
f(x) = q(x)·g(x) + r(x),
with deg r < deg g (or r = 0).

- Each step: look at the leading term of the current remainder and the leading term of g(x). Divide those leading terms to get a monomial m(x) (this division of coefficients requires working over a field or a ring where the leading-coefficient division is possible). Multiply g(x) by m(x) (polynomial multiplication) and subtract that product from the current remainder (polynomial subtraction = addition of a polynomial with the negative of another). The new remainder is again a polynomial.

- Repeat until the remainder has smaller degree than g(x). The quotient q(x) is the sum of the monomials m(x) chosen at each step. Each m(x) is a polynomial, and sums of polynomials are polynomials. Each product m(x)·g(x) is a polynomial, and each subtraction produces a polynomial.

Because every intermediate result—the partial quotients, the products, the successive remainders—comes from multiplying and adding polynomials, the algorithm never leaves the set of polynomials. The final q(x) and r(x) are therefore polynomials. That is a concrete verification that polynomials are closed under multiplication and addition: the division process only uses those operations and produces polynomials at every stage.

(As a side remark: to guarantee you can divide leading coefficients in the step "divide leading terms" you normally work over a field of coefficients. Closure under addition and multiplication, however, holds for polynomial rings over any coefficient ring.)