Mathematics : Theory of Equations: Polynomial Equations with no Additional Information: Non-polynomial Equations

**Non-polynomial Equations**

Some non-polynomial equations can be solved using polynomial
equations. As an example let us consider the equation âˆš(15-2*x*) = *x. *First
we note that this is not a polynomial equation. Squaring both sides, we get *x*^{2} + 2*x *âˆ’15 = 0 . We know how to
solve this polynomial equation. From the solutions of the polynomial equation,
we can analyse the given equation. Clearly 3 and âˆ’5 are solutions of *x*^{2} + 2*x *âˆ’15 = 0 . If we adopt the
notion of assigning only nonnegative values for âˆš â€¢ then *x *= 3 is the only
solution; if we do not adopt the notion, then we get *x *= âˆ’5 is also a solution.

Find solution, if any, of the equation 2 cos2 *x *âˆ’ 9 cos *x
*+ 4 = 0

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 (1) by using our knowledge on
polynomial equations. If we replace cos *x *by *y *, then we get the
polynomial equation 2y^{2} - 9y + 4 = 0 for which 4 and 1/2 are solutions.

From this we conclude that *x *must satisfy cos *x *=
4 or cos *x *= 1/2. But cos *x *= 4 is never possible, if we take cos
*x *= 1/2 , then we get infinitely many real numbers *x *satisfying
cos *x *= 1/2 ; in fact, for all are solutions for the
given equation (1).

If we repeat the steps by taking the equation cos^{2}*x *- 9 cos *x *+ 20
= 0, we observe that this equation
has no solution.

We note that

â€¢ not all solutions of the derived polynomial equation give a
solution for the given equation;

â€¢ there may be infinitely many solutions for non-polynomial
equations though they look like polynomial equations;

â€¢ there may be no solution for such equations.

â€¢ 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.

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

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12th Mathematics : UNIT 3 : Theory of Equations : Non-polynomial Equations | Solved Example Problems | Theory of Equations

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