- - For instance, the equation x has only one irrational root, namely . Of course, the other two roots are imaginary numbers (What are they?). Example . Find a polynomial equation of minimum degree with rational coefficients, having as a root.
Since is a root and the coefficients are rational numbers, is also a root. A required polynomial equation is given by x − (Sum of the roots) x + Product of the roots = and hence + = is a required equation. Note We note that the term “rational coefficients” is very important; otherwise, x − 0will be a polynomial equation which has as a root but not . We state the following result without proof.
Theorem . Let p and q be rational numbers so that p and q are irrational numbers; further let one of p and q be not a rational multiple of the other. If q ++ is a root of a polynomial equation with rational coefficients, then q q , and − q are also roots of the same polynomial equation. Example .
Form a polynomial equation with integer coefficients with as a root. Since is a root, x − is a factor. To remove the outermost square root, we take x + as another factor and find their product = x Still we didn’t achieve our goal. So we include another factor x and get the product = x So, x − is a required polynomial equation with the integer coefficients.
Now we identify the nature of roots of the given equation without solving the equation. The idea comes from the negativity, equal to and positivity of ∆= ac - - Theory of Equations