Answer :
To determine the characteristic of the polynomial's order when it has only one real non-repeated root, let's analyze the fundamental properties of polynomials and their roots.
### Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that every non-zero polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots when counted with multiplicity. These roots may be real or complex.
### Roots and Polynomial Degree
If a polynomial has only one real root and this root is not repeated (i.e., its multiplicity is 1), we can infer specific characteristics about the polynomial:
1. Even-Degree Polynomials: An even-degree polynomial, such as a quadratic (degree 2) or quartic (degree 4) polynomial, always has an even number of roots due to the Fundamental Theorem of Algebra. For even-degree polynomials, complex roots come in conjugate pairs. This implies that if a polynomial has real roots, they too must come in pairs.
2. Odd-Degree Polynomials: An odd-degree polynomial, such as a cubic (degree 3) or quintic (degree 5) polynomial, always has an odd number of roots.
### Conclusion for Only One Real Non-Repeated Root
Given the properties above:
- If a polynomial has only one real non-repeated root, it implies the total number of roots which must include complex roots if they exist.
- An even-degree polynomial cannot have only one real root because it would contradict the necessary paired structure of real and complex roots.
- Therefore, the polynomial must be of odd degree, as only odd-degree polynomials can have exactly one real root (with or without additional complex roots).
Following this logical reasoning, we conclude that:
If a polynomial has only one real non-repeated root, the polynomial must be of odd degree.
### Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that every non-zero polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots when counted with multiplicity. These roots may be real or complex.
### Roots and Polynomial Degree
If a polynomial has only one real root and this root is not repeated (i.e., its multiplicity is 1), we can infer specific characteristics about the polynomial:
1. Even-Degree Polynomials: An even-degree polynomial, such as a quadratic (degree 2) or quartic (degree 4) polynomial, always has an even number of roots due to the Fundamental Theorem of Algebra. For even-degree polynomials, complex roots come in conjugate pairs. This implies that if a polynomial has real roots, they too must come in pairs.
2. Odd-Degree Polynomials: An odd-degree polynomial, such as a cubic (degree 3) or quintic (degree 5) polynomial, always has an odd number of roots.
### Conclusion for Only One Real Non-Repeated Root
Given the properties above:
- If a polynomial has only one real non-repeated root, it implies the total number of roots which must include complex roots if they exist.
- An even-degree polynomial cannot have only one real root because it would contradict the necessary paired structure of real and complex roots.
- Therefore, the polynomial must be of odd degree, as only odd-degree polynomials can have exactly one real root (with or without additional complex roots).
Following this logical reasoning, we conclude that:
If a polynomial has only one real non-repeated root, the polynomial must be of odd degree.
