Alright, let's determine the number of possible imaginary (complex) solutions for the given polynomial equation. The polynomial equation given is:
[tex]\[ x^4 - 7x^3 - x^2 + 67x - 60 = 0 \][/tex]
To determine the number of imaginary solutions for this polynomial, we will follow these steps:
1. Understanding the Polynomial:
- This is a fourth-degree polynomial, meaning it can have up to 4 solutions. The nature of these solutions can vary—they can be real or complex (imaginary).
2. Fundamental Theorem of Algebra:
- According to the Fundamental Theorem of Algebra, a polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots, considering multiplicity and the possibility of complex roots.
3. Complex Roots and Conjugate Pairs:
- Complex roots, if they exist, occur in conjugate pairs for polynomials with real coefficients. So, if there are any complex roots, they must appear in pairs.
4. Nature of Roots in This Polynomial:
- After thoroughly evaluating the polynomial, it is determined that all roots are real. There are four real roots and, as a result, no imaginary roots.
5. Conclusion:
- Since the number of complex (imaginary) roots is zero, we conclude there are no imaginary solutions for this polynomial.
Thus, the number of possible imaginary (complex) solutions for the polynomial [tex]\( x^4 - 7x^3 - x^2 + 67x - 60 = 0 \)[/tex] is:
[tex]\[ \boxed{0} \][/tex]
