Nature of Roots and Nature of Coefficients of Polynomial Equations

Rational Roots

If all the coefficients of a quadratic equation are integers, then Î” is an integer, and when it is positive, we have, âˆšÎ” is rational if, and only if, Î” is a perfect square. In other words, the equation *ax*2 + *bx *+ *c *= 0 with integer coefficients has rational roots, if, and only if, Î” 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 1 with rational coefficients having 1/2 as a root, whereas there is no monic polynomial equation of any degree with integer coefficients having 1/2 as a root.

Show that the equation 2*x*2 - 6*x *+ 7 = 0 cannot be satisfied by any real values of *x.*

âˆ†= *b*2 âˆ’ 4*ac *= âˆ’20 < 0 . The roots are imaginary numbers.

Example 3.12

If *x*2 + 2 (*k *+ 2)*x *+ 9*k *= 0 has equal roots, find *k*.

Solution

Here Î” = *b*2 - 4*ac *= 0 for equal roots. This implies 4 (*k *+ 2)2 = 4 (9) *k *.This implies *k *= 4 or 1.

Example 3.13

Show that, if *p*, *q*, *r* are rational, the roots of the equation *x*2 - 2 *px *+ *p*2 - *q*2 + 2*qr *- *r*2 = 0 are rational.

Solution

The roots are rational if Î” = *b*2 - 4*ac *= (-2 *p*)2 - 4 ( *p*2 - *q*2 + 2*qr *- *r*2 ) .

But this expression reduces to 4 (*q*2 - 2*qr *+ *r*2 ) or 4 (*q *- *r *)2 which is a perfect square. Hence the roots are rational.

Tags : Definition, Solved Example Problems | Theory of Equations , 12th Mathematics : UNIT 3 : Theory of Equations

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12th Mathematics : UNIT 3 : Theory of Equations : Rational Roots | Definition, Solved Example Problems | Theory of Equations

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