Question

how many zeros does the fundamental theorem of algebra guarantee that this polynomial function will have? f, [, x, ], equals, x, to the power 5 , plus, 2, x, to the power 4 , minus, 5, x, squared, plus, 2, x, plus, 1f(x)

Answer 1

❣.•° <

< °•.❣

Explanation:

The Fundamental Theorem of Algebra states that every non-constant polynomial equation with complex coefficients has at least one complex root. In the context of counting zeros (or roots) for a polynomial, the theorem implies that a polynomial of degree \( n \) has exactly \( n \) complex roots when counted with multiplicity.

For the given polynomial function \( f(x) = x^5 + 2x^4 - 5x^2 + 2x + 1 \), the highest power of \( x \) is 5. Therefore, according to the Fundamental Theorem of Algebra, this polynomial has 5 complex roots.

It's important to note that some of these roots may be repeated, and they could be real or complex. The theorem doesn't specify the nature of the roots, only their count.