Graph the polynomial on a graphing calculator. The number of times the graphed curve crosses the X axis is the number of roots you can find by factoring. If the number of X-axis crossings is the same as the degree --- the size of largest exponent --- of the polynomial, the polynomial can be factored into monomials, or simple expressions with no exponents. For example, the graph of 2X^3 -11X^2 + 19X -10 crosses the X axis three times, so it can be factored into three monomials.
Factor a polynomial 2X^3 -11X^2 + 19X -10 by considering the cofficient of the leading term and the factors of the constant term. The cofficient of the leading term is 2, which has factors 1 and 2, and the constant term is 10, which has factors 1, 2, 5 and 10. These factors generate the candidates for factors: X - 1, X + 1, X - 2, X + 2, X - 5, X + 5, X - 10, X + 10, 2X - 1, 2X + 1, 2X - 2, 2X + 2, 2X - 5, 2X + 5, 2X - 10 and 2X + 10. Try dividing each of these into 2X^3 -11X^2 + 19X -10 to find the monomial divisors of the polynomial.
Trying all of the candidate factors reveals that X - 1, X - 2 and 2X - 5 all divide 2X^3 -11X^2 + 19X -10. The degree is three, and we have found three factors, so 2X^3 -11X^2 + 19X -10 = (X - 1)(X - 2)(2X - 5). Here is an example of a polynomial that does not have all monomial factors: Z^3 + 3Z^2 +3Z + 2 has candidate divisors Z - 1, Z + 1, Z - 2 and Z + 2, but only Z + 2 divides the polynomial. So Z^3 + 3Z^2 +3Z + 2 = (Z + 2)(Z^2 + Z + 1), because Z^2 + Z + 1 can not be factored. The candidate divisors of Z^2 + Z + 1 are Z -1 and Z + 1 and neither divide the polynomial.