Question

fill in the blank so that the resulting statement is true.

if a polynomial equation is of degree n, then counting multiple roots separately, the equation has roots.

options:
n - 1
n
n + 1

Explanation:

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<reasoning>

Identify the mathematical theorem

Using the Polynomial Functions and Degree of Polynomial knowledge points
The statement describes the Fundamental Theorem of Algebra, which relates the degree of a polynomial equation to its total number of complex roots.

Determine the number of roots

Using the Polynomial Functions and Degree of Polynomial knowledge points
A polynomial equation of degree \(n\) (where \(n \ge 1\)) has exactly \(n\) roots in the complex number system, provided that multiple roots (roots with multiplicity greater than 1) are counted separately according to their multiplicity.
</reasoning>

<answer>
If a polynomial equation is of degree n, then counting multiple roots separately, the equation has <blank>\(n\)</blank> roots.
</answer>

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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Fundamental Theorem of Algebra"
]
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Answer:

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<reasoning>

Identify the mathematical theorem

Using the Polynomial Functions and Degree of Polynomial knowledge points
The statement describes the Fundamental Theorem of Algebra, which relates the degree of a polynomial equation to its total number of complex roots.

Determine the number of roots

Using the Polynomial Functions and Degree of Polynomial knowledge points
A polynomial equation of degree \(n\) (where \(n \ge 1\)) has exactly \(n\) roots in the complex number system, provided that multiple roots (roots with multiplicity greater than 1) are counted separately according to their multiplicity.
</reasoning>

<answer>
If a polynomial equation is of degree n, then counting multiple roots separately, the equation has <blank>\(n\)</blank> roots.
</answer>

<post_analysis>
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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Fundamental Theorem of Algebra"
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