. . Non-polynomial Equations Some non-polynomial equations can be solved using polynomial equations. As an example let us consider the equation x .
First we note that this is not a polynomial equation. Squaring both sides, we get x . We know how to solve this polynomial equation. From the solutions of the polynomial equation, we can analyse the given equation.
Clearly and − are solutions of x . If we adopt the notion of assigning only nonnegative values for · then x = is the only solution; if we do not adopt the notion, then we get x = − is also a solution. Example . : Find solution, if any, of the equation = .
... ( ) The left hand side of this equation is not a polynomial in x . But it looks like a polynomial. In fact, we can say that this is a polynomial in cos x .
However, we can solve equation ( ) by using our knowledge on polynomial equations. If we replace cos x by y , then we get the polynomial equation y for which and are solutions. From this we conclude that x must satisfy cos x = 4orcos x = . But cos x = is never possible, if we take cos x = , then we get infinitely many real numbers x satisfying cos x = ; in fact, for all n ∈ , x ± π are solutions for the given equation ( ).
If we repeat the steps by taking the equation cos we observe that this equation has no solution. - - Remarks We note that • not all solutions of the derived polynomial equation give a solution for the given equation; • there may be infinitely many solutions for non-polynomial equations though they look like polynomial equations; • there may be no solution for such equations. • the Fundamental Theorem of Algebra is proved only for polynomials; for non-polynomial expressions, we cannot talk about degree and hence we should not have any confusion on the Fundamental Theorem of Algebra having non-polynomial equations in mind.