. . Rational Root Theorem We can find a few roots of some polynomial equations by trial and error method. For instance, we consider the equation = ...
( ) This is a third degree equation which cannot be solved by any method so far we discussed in this chapter. If we denote the polynomial in ( ) as P x ( ), then we see that P ( ) which says that x − is a factor. As the rest of the problem of solving the equation is easy, we leave it as an exercise. - - Theory of Equations Example .
Solve the equation x If P x ( ) denotes the polynomial in the equation, then P ( ) . Hence is a root of the polynomial equations. To find other roots, we divide the given polynomial x by x − and get Q ( x )= x as the quotient. Solving Q ( x ) = we get − and as roots.
Thus are the solutions of the given equation. Guessing a number as a root by trial and error method is not an easy task. But when the coefficients are integers, using its leading coefficient and the constant term, we can list certain rational numbers as possible roots. Rational Root Theorem helps us to create such a list of possible rational roots.
We recall that if a polynomial has rational coefficients, then by multiplying by suitable numbers we can obtain a polynomial with integer coefficients having the same roots. So we can use Rational Root Theorem , given below, to guess a few roots of polynomial with rational coefficient. We state the theorem without proof. Theorem .
(Rational Root Theorem) Let a x a x n + with a n ¹ and a ¹ , be a polynomial with integer coefficients. If q , with ( , ) p q = , is a root of the polynomial, then p is a factor of a and q is