. . Rational Roots If all the coefficients of a quadratic equation are integers, then D is an integer, and when it is positive, we have, D is rational if, and only if, D is a perfect square. In other words, the equation ax bx with integer coefficients has rational roots, if, and only if, D is a perfect square.
What we discussed so far on polynomial equations of rational coefficients holds for polynomial equations with integer coefficients as well. In fact, multiplying the polynomial equation with rational coefficients, by a common multiple of the denominators of the coefficients, we get a polynomial equation of integer coefficients having the same roots. Of course, we have to handle this situation carefully. For instance, there is a monic polynomial equation of degree with rational coefficients having as a root, whereas there is no monic polynomial equation of any degree with integer coefficients having as a root.
Example . Show that the equation x cannot be satisfied by any real values of x. ∆= =− < ac . The roots are imaginary numbers.
Example . If x ) + has equal roots, find k . Here ∆= ac for equal roots. This implies ) = ( ) .This implies k = or .
Example . Show that, if p q r , , are rational, the roots of the equation x px q qr are rational. The roots are rational if ∆= ) − ac q qr But this expression reduces to q qr