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

Chapter: 12th Mathematics : UNIT 3 : Theory of Equations

Rational Roots

Nature of Roots and Nature of Coefficients of Polynomial Equations

Rational Roots

If all the coefficients of a quadratic equation are integers, then Δ is an integer, and when it is positive, we have, √Δ is rational if, and only if,  Δ   is a perfect square. In other words, the equation   ax2 + bx + = 0 with integer coefficients has rational roots, if, and only if, Δ 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.

 

Example 3.11

Show that the equation 2x2 - 6+ 7 = 0 cannot be satisfied by any real values of x.

Solution

∆= b2 − 4ac = −20 < 0 . The roots are imaginary numbers.

 

Example 3.12

If x2 + 2 (+ 2)+ 9= 0 has equal roots, find k.

Solution

Here Δ = b2  - 4ac = 0  for equal roots. This implies 4 (+ 2)2  = 4 (9) .This implies = 4 or 1.

 

Example 3.13

Show that, if pqr are rational, the roots of the equation x2 - 2 px p2 - q2 + 2qr r2 = 0 are rational.

Solution

The roots are rational if  Δ = b2  - 4ac = (-2 p)2  - 4 ( p2  - q2  + 2qr r2 ) .

But this expression reduces to 4 (q2  - 2qr r2 ) or 4 ()2   which is a perfect square.  Hence the roots are rational.


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12th Mathematics : UNIT 3 : Theory of Equations


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