Find the rational roots by first finding all the factors of the first and last numbers in the polynomial. Write each factor in the second set over each number in the first set -- with both plus and minus signs. For example, for the polynomial 2X^2 - 7X + 6, the first number is 2 -- which has factors 1 and 2. The last number is 6, which has factors 1, 2, 3 and 6. The rational roots are +1/1, -1/1, +2/1, -2/1, +3/1, -3/1, +6/1, -6/1, +1/2, -1/2, +2/2, -2/2, +3/2, -3/2, +6/2 and -6/2 or 1, -1, 2, -2, 3, -3, 6, -6, 1/2, -1/2, 3/2 and -3/2.
The maximum number of zeros of a polynomial is given by the value of the largest exponent in the polynomial. Therefore, 2X^2 - 7X + 6 has at most two roots. If two of the rational roots are zeros, the job is done. Graphing the polynomial on a graphing calculator will save a lot of time and also give some indication of what kind of roots the polynomial has. If the graphed curve crosses the X axis at any of the rational roots, try these rational roots first; they are likely the correct ones. If there are fewer X axis crossings than roots, some of the roots may be complex.
Some polynomials have irrational real roots. For example, X^2 -2 has two roots -- the positive and negative square roots of 2. The graphed curve will cross the X axis at +1.414 and -1.414. This might be difficult to distinguish from +3/2 and -3/2, which underscores the importance of checking your answers computationally instead of just reading the value off the graph.
If some of the roots are missing -- especially if there is an even number of roots missing -- the polynomial may have complex roots. If the polynomial is a quadratic, you can calculate the roots using the quadratic formula, which will give the complex roots. If some of the roots were rational, divide the original polynomial by X - r, where "r" is a root. If the quotient is a quadratic, use the quadratic formula. Complex roots come in pairs, so if you are missing only one root, the missing root is not complex.