Let's analyze the given polynomial function:
[tex]\[ f(x) = 8x^7 - x^5 + x^3 + 6 \][/tex]
First, let's recall the Fundamental Theorem of Algebra. This theorem states that every non-zero single-variable polynomial with complex coefficients has as many roots as its degree, counting multiplicities. So, if we can determine the degree of the polynomial, we can determine the number of roots it possesses.
The degree of a polynomial is the highest power of the variable [tex]\( x \)[/tex] that appears in the polynomial with a non-zero coefficient.
For the polynomial [tex]\( f(x) = 8x^7 - x^5 + x^3 + 6 \)[/tex]:
- The term [tex]\( 8x^7 \)[/tex] has the exponent 7.
- The term [tex]\( -x^5 \)[/tex] has the exponent 5.
- The term [tex]\( x^3 \)[/tex] has the exponent 3.
- The term [tex]\( 6 \)[/tex] is a constant and can be considered as having the exponent 0.
Among these, the highest exponent is 7, which is the degree of the polynomial.
According to the Fundamental Theorem of Algebra, a polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots (counting multiplicities).
Since the degree of the given polynomial [tex]\( f(x) = 8x^7 - x^5 + x^3 + 6 \)[/tex] is 7, it must have exactly 7 roots.
Therefore, the correct number of roots for the polynomial function is:
[tex]\[ \boxed{7} \][/tex]
