Question

[100 POINTS!!!]

According to the Fundamental Theorem of Algebra, how many zeros does the function f(x) = 5x^6+2x^3−4x + 1


There are____________ Zeros.

Answer 1

The Fundamental Theorem of Algebra states that every non-constant polynomial function has at least one complex root. In the case of the polynomial function f(x) = 5x^6+2x^3−4x + 1, it is a sixth-degree polynomial function, which means it has at most six complex roots. However, the Fundamental Theorem of Algebra does not tell us the exact number of roots, nor does it tell us whether the roots are real or complex.

To determine the number of real roots of the function f(x), we can use the Intermediate Value Theorem or graph the function to find the number of times it crosses the x-axis. However, determining the number of complex roots requires more advanced techniques, such as the use of the Fundamental Theorem of Algebra in combination with the Factor Theorem, Rational Root Theorem, or other methods.

Therefore, based on the Fundamental Theorem of Algebra alone, we can say that the function f(x) = 5x^6+2x^3−4x + 1 has at least one complex root, but we cannot determine the exact number or nature of the roots without additional analysis.

Answer 2

Explanation:

According to the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n complex zeros (counting multiplicities).

The degree of the polynomial f(x) = 5x^6+2x^3−4x + 1 is 6, which means that it is a sixth-degree polynomial. Therefore, it has exactly 6 complex zeros (counting multiplicities).

Note that the zeros may be real or complex, and they may not all be distinct. To find the exact number of zeros and their values, we would need to use additional methods such as factoring, the rational zeros theorem, or numerical methods.