Question

Which is a possible combination of root types for a 7th degree polynomial?

Which is a possible combination of root types for a 7th degree polynomial?
6 rational roots, 1 irrational root
3 irrational roots, 4 complex roots
2 rational roots, 2 irrational roots, 3 complex roots
7 rational roots

Answer 1

Given:

Degree of a polynomial is 7.

To find:

The possible combination of root types for a 7th degree polynomial.

Solution:

We know that, by complex conjugate root theorem, is a complex number is a root of a polynomial, then its conjugate is also the root of that polynomial, it means number of complex roots always an even number.

Similarly,

Irrational roots are also occurs in pairs. If
`1+√(2)` is a root of polynomial, then its
`1-√(2)` is also the root of that polynomial, it means number of irrational roots always an even number.

In options A, B and C either complex or irrational roots are odd, which is not true.

Therefore, the correct option is D.