Question
A polynomial equation includes the term 8x5
8
𝑥
5
. 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.
8
𝑥
5
. 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.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
According to the Fundamental Theorem of Algebra, a polynomial of degree \( n \) has exactly \( n \) roots, counting multiplicities, in the complex number system. The term \( 8x^5 \) indicates that the polynomial is of degree 5.
Therefore, the correct statement is:
**The equation has at least 5 roots.**
This means that the polynomial must have exactly 5 roots (some of them may be complex or repeated). Hence, it's not guaranteed that all roots are real, nor does it imply an odd number of real roots specifically.
Therefore, the correct statement is:
**The equation has at least 5 roots.**
This means that the polynomial must have exactly 5 roots (some of them may be complex or repeated). Hence, it's not guaranteed that all roots are real, nor does it imply an odd number of real roots specifically.
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