To determine the number of zeros of the polynomial [tex]\( f(x) = x^5 - 12x^3 + 7x - 5 \)[/tex], we can apply the Fundamental Theorem of Algebra.
Step-by-Step Solution:
1. Identify the degree of the polynomial:
The degree of a polynomial is the highest power of [tex]\( x \)[/tex] with a non-zero coefficient in the polynomial.
In the given polynomial [tex]\( f(x) = x^5 - 12x^3 + 7x - 5 \)[/tex], the highest power of [tex]\( x \)[/tex] is 5.
2. Apply the Fundamental Theorem of Algebra:
The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has exactly as many complex roots (including multiplicity) as its degree.
3. Conclusion:
Since the degree of the polynomial [tex]\( f(x) = x^5 - 12x^3 + 7x - 5 \)[/tex] is 5, it follows that this polynomial has exactly 5 zeros.
Therefore, the polynomial [tex]\( f(x) = x^5 - 12x^3 + 7x - 5 \)[/tex] has exactly 5 zeros.
Answer:
5
