How to Find the Roots of a Polynomial Equation

By Carlos Mano
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Polynomials are mathematical expressions formed by adding terms of a variable to different powers. The largest exponent -- the degree of the polynomial -- tells you the maximum number of roots the polynomial can have. A root of a polynomial is a number that, if it is substituted in for the unknown variable in the polynomial, the resulting expression would evaluate to zero. Finding the roots involves finding factors of the polynomial -- simpler expressions that can be multiplied together to give the polynomial.

Set the polynomial to a variable such as Y = X^3 + 3X^2 - X - 3, and graph the function. The places where the graphed curve crosses the X axis indicates the roots of the polynomial. The calculator cannot give the roots accurately, but it can indicate how many times the curve crosses the X axis. If the number of X axis crossings are the same as the largest exponent in the polynomial, then the polynomial can be factored into monomials -- expressions without exponents. These simple expressions are easy to solve for the exact roots. For example, the graph of Y = X^3 + 3X^2 - X - 3 crosses the X axis three times, so it can be factored into three monomials.

Factor the polynomial by considering candidates generated by factors of the constant term. In X^3 + 3X^2 - X - 3, the constant term is 3, which has factors 1 and 3. The candidates for factors are X - 1, X + 1, X - 3 and X + 3. Trying each of these you will find that X - 1, X + 1 and X + 3 all divide X^3 + 3X^2 - X - 3 In other words, X^3 + 3X^2 - X - 3 = (X - 1)(X + 1)(X + 3).

Set each of the factors of a polynomial to zero and solve to get one of the roots. For the polynomial X^3 + 3X^2 - X - 3, the factors are X - 1, X + 1 and X + 3. Solving the three equations X -1 = 0, X + 1 = 0 and X + 3 = 0, gives the three roots: -1, + 1 and -3. Substituting either of these roots into the polynomial will evaluate to zero. So (-1)^3 + 3(-1)^2 - (-1) - 3 = 0 and (+1)^3 + 3(+1)^2 - (+1) - 3 = 0 and (-3)^3 + 3(-3)^2 - (-3) - 3 = 0 and these are the only numbers that will make the polynomial evaluate to zero.