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

There are four possible combinations of root types for a 7th degree polynomial:

(i) 7 rational roots. (, , )

(ii) 6 rational roots, 1 irrational root. (, , )

(iii) 2 rational roots, 1 irrational root, 4 complex roots. (, , )

(iv) 3 irrational roots, 4 complex roots. (, , )

Step-by-step explanation:

According to the Fundamental Theorem of Algebra, every polynomial with degree higher than zero has at least a solution and at best complex roots, where is the degree of the polynomial. In this case, we have 7 roots.

We can rewrite the 7th degree polynomial as a product of a 3rd degree polynomial and 4th degree polynomial. That is:

(Eq. 1)

From Algebra we know that both 3rd degree and 4th degree polynomials are solvable by analytical means:

1) All 4th degree polynomials are solvable by Ferrari's method, whose posible solutions are, respectively:

(i) 4 rational roots.

(ii) 4 irrational roots.

(iii) 2 rational roots, 2 irrational roots.

(iv) 3 rational roots, 1 irrational root.

(v) 1 rational roots, 3 irrational roots.

(vi) 2 rational roots, 2 complex roots.

(vii) 2 irrational roots, 2 complex roots.

(viii) 1 rational root, 1 irrational root, 2 complex roots.

(ix) 4 complex roots.

2) All 3rd degree polynomials are solvable by Cardano's method, whose possible solutions are, respectively:

(i) 3 rational roots.

(ii) 3 irrational roots.

(iii) 1 rational root, 2 complex roots.

(iv) 1 irrational root, 2 complex roots.

(v) 3 complex roots.

Please note that complex roots are always presented in conjugated pairs by Quadratic Formula.

The number of solutions of a polynomials are represented by the following expression:

(Eq. 2)

Where:

- Number of rational roots, dimensionless.

- Number of irrational roots, dimensionless.

- Number of pairs of complex roots, dimensionless.

Based on this expression, we find the possible root types for a 7th degree polynomial:

(i) 7 rational roots. (, , )

(ii) 6 rational roots, 1 irrational root. (, , )

(iii) 2 rational roots, 1 irrational root, 4 complex roots. (, , )

(iv) 3 irrational roots, 4 complex roots. (, , )

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