Final answer:
The equation 3x^5 + 10x^4 - x^2 + 6x + 9 = 0, which is a quintic equation, will have exactly 5 roots, per the Fundamental Theorem of Algebra. These roots can be real or complex and while finding exact roots for high-degree polynomials may be difficult, numerical methods can be used.
Step-by-step explanation:
To determine the number of roots for the equation 3x^5 + 10x^4 - x^2 + 6x + 9 = 0, we can apply the Fundamental Theorem of Algebra which states that every non-zero, single-variable, degree n polynomial with complex coefficients has exactly n roots, counting multiplicity. For the equation given, the highest degree is 5, meaning that the polynomial is a quintic equation, and it will have exactly 5 roots. These roots could be real or complex numbers.
It is not always straightforward to find the exact roots of a polynomial equation of degree higher than 4, such as this one, by algebraic methods alone. However, numerical methods or a graphing calculator could be used to approximate the roots if required. For a quadratic equation, like ax^2 + bx + c = 0, the number of roots can be calculated using the quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a), which always yields two roots (which can be real or complex depending on the discriminant b^2 - 4ac).