Reducing a polynomial (over the field of integers) as the product of polynomials with smaller degree is one of the basic tasks taught in any algebra class. However, polynomials seen in textbooks are rarely more complicated than a quadratic (of degree = 2) or a cubic (of degree = 3). Many methods of reducting polynomials of higher degrees usually depend on somehow finding a linear factor (a root), dividing, and then dealing with the remainder. But what if a given polynomial doesn't have a simple, linear factor?

The thesis will uncover a method of reducing a polynomial (f(x)) with non-negative coefficients (all coefficients greater than or equal to zero) by considering the prime-power decomposition of the value of f(10). These prime factors will produce the candidates for the possible factors of f(x).

A brief preview: Let f(x) = 3x^7 + 4x^6 + 9x^4 + 5x^3 + 2x^2 + 8x + 9. f(10) = 34095289, which is a prime number. Hence f(x) is irreducible. And, in fact, it is always the case thet if f(10) is a prime number, then f(x) is irreducible (provided all the coefficients are less than 1,000,000,000,000,000).