. . Imaginary or Surds Roots If α β + i is an imaginary root of a quartic polynomial with real coefficients, then α β − i is also a root; thus ( )) β and( )) β are factors of the polynomial; hence their product is a factor; in other words, x β is a factor; we can divide the polynomial with this factor and get the second degree quotient which can be solved by known techniques; using this we can find all the roots of the polynomial. - - If is a root of a quadric polynomial equation with rational coefficients, then is also a root; thus their product ( ))( )) is a factor; that is x + is a factor; we can divide the polynomial with this factor and get the quotient as a second degree factor which can be solved by known techniques.
Using this, we can find all the roots of the quadric equation. This technique is applicable for all surds taken in place of If an imaginary root and a surd root of a sixth degree polynomial with rational coefficient are known, then step by step we may reduce the problem of solving the sixth degree polynomial equation into a problem of solving a quadratic equation. Example . If + i and are roots of the equation find all roots.
Since the coefficient of the equations are all rational numbers, and + i and3 are roots, we get − i and are also roots of the given equation. Thus( )), )),( )) and ( )) x − are factors. Thus their product (( ))( ))( ))( )) is a factor of the given polynomial equation. That is, )( is a factor.
Dividing the given polynomial equation by this factor, we get the other factor as ( which implies that and − are the other two roots. Thus , and are the roots of the given polynomial equation.