and can be proved using the technique used in the proof of imaginary roots, we state and prove this only for a quadratic equation in Theorem . . Before proving the theorem, we recall that if a and b are rational numbers and c is an irrational number such that a bc is a rational number, then b must be ; further if a bc = , then a and b must be . For instance, if a ∈ , then b must be , and if a then a = .
Now we state and prove a general result as given below. Theorem . Let p and q be rational numbers such that q is irrational. If p q is a root of a quadratic equation with rational coefficients, then p q is also a root of the same equation.
Proof We prove the theorem by assuming that the quadratic equation is a monic polynomial equation. The result for non-monic polynomial equation can be proved in a similar way. Let p and q be rational numbers such that q is irrational. Let p q be a root of the equation bx where b and c are rational numbers.
Let α be the other root. Computing the sum of the roots, we get α + q = − b and hence α + = −− ∈ q . Taking −− p as s , we have α + q s . This implies that α = s q Computing the product of the roots, we get )( s q q = c and hence( sp q s q = ∈ .
Thus s = . This implies that s and hence we get α = q . So, the other root is p q Remark The statement of Theorem . may seem to be a little bit complicated.
We should not be in a hurry to make the theorem short by writing “ for a polynomial equation with rational coefficients, irrational roots occur in pairs ” . This is not true.