Polynomials are akin to integers in that they are closed under addition, subtraction, and multiplication operations. The Remainder Theorem holds that for any polynomial p(x) and any number a, the division of p(x) by x - a leaves the remainder p(a), hence p(a) equals 0 just when (x - a) is a factor of p(x). It is possible to locate the zeros of polynomials when suitable factorizations can be performed, and these zeros aid in the approximate plotting of the function defined by the polynomial.

Proving polynomial identities enables the description of number relationships, as shown in the example of the polynomial identity (x2 + y2)2 = (x2 - y2)2 + (2xy)2 for generating Pythagorean triples. The Binomial Theorem can be applied to the expansion of (x + y)n as powers of x and y where x and y can be any numbers, and n is a positive integer.

Simple rational expressions can be transcoded into different forms, including putting a(x)/b(x) in the form q(x) + r(x)/b(x), where each of a(x), b(x), q(x), and r(x)