A polynomial equation includes the term 8x5 . According to the Fundamental Theorem of Algebra, which statement must be true?(1 point)

Responses

The equation has more than 5 roots.
The equation has more than 5 roots.

The equation has at least 5 roots.
The equation has at least 5 roots.

The equation has an odd number of real roots.
The equation has an odd number of real roots.

All the roots of the equation are real.

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
6 days ago

According to the Fundamental Theorem of Algebra, a polynomial of degree \( n \) has exactly \( n \) roots, counting multiplicities. In the case of the polynomial equation with the term \( 8x^5 \), the degree of the polynomial is 5.

Based on this information, the correct statement is:

**The equation has at least 5 roots.**

This statement is true because a polynomial of degree 5 must have exactly 5 roots (which may include complex roots and multiplicities, but it must have 5 roots in total).

The other statements are not necessarily true:
- The equation does not have more than 5 roots; it has exactly 5.
- The statement about having an odd number of real roots is not guaranteed; there could be complex roots.
- Not all roots must be real; it can have complex roots as well.

So, the answer you should select is:

**The equation has at least 5 roots.**