3.7.2 Polynomial equations with Even Powers Only

. . Polynomial equations with Even Powers Only If P x ( ) is a polynomial equation of degree n , having only even powers of x , (that is, coefficients of odd powers are ) then by replacing x by y , we get a polynomial equation with degree n in y ; let y y y n ,  be the roots of this polynomial equation. Then considering the n equations x y r = , we can find two values for x for each y r ; these n numbers are the roots of the given polynomial equation in x .

Example . Solve the equation x The given equation is This is a fourth degree equation. If we replace x by y , then we get the quadratic equation Theory of Equations It is easy to see that and as solutions for y . Now taking x and x we get as solutions of the given equation.

We note that the technique adopted above can be applied to polynomial equations like , ax bx and in general polynomial equations of the form a x a x kn k n  where k is any positive integer.