Question
Apply the Fundamental Theorem of Algebra to find the number of roots for the polynomial g(x)=7x^6+2x−5 .
Answers
GPT-4o mini
The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) has exactly \( n \) roots, counting multiplicities, in the complex number system.
For the polynomial \( g(x) = 7x^6 + 2x - 5 \):
1. **Determine the degree of the polynomial:** The highest power of \( x \) in \( g(x) \) is \( 6 \) (from the term \( 7x^6 \)).
2. **Apply the Fundamental Theorem of Algebra:** Since the degree of the polynomial is \( 6 \), the polynomial \( g(x) \) has exactly \( 6 \) roots in the complex number system.
Thus, the polynomial \( g(x) = 7x^6 + 2x - 5 \) has **6 roots** in total.
For the polynomial \( g(x) = 7x^6 + 2x - 5 \):
1. **Determine the degree of the polynomial:** The highest power of \( x \) in \( g(x) \) is \( 6 \) (from the term \( 7x^6 \)).
2. **Apply the Fundamental Theorem of Algebra:** Since the degree of the polynomial is \( 6 \), the polynomial \( g(x) \) has exactly \( 6 \) roots in the complex number system.
Thus, the polynomial \( g(x) = 7x^6 + 2x - 5 \) has **6 roots** in total.