Question

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.

Answer 1

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.**