- The polynomial \(ax^2 + bx + c\) with \(a
eq 0\) is a quadratic polynomial of degree 2.
- The Fundamental Theorem of Algebra states that a polynomial of degree \(n\) has exactly \(n\) complex roots, counting multiplicities.
- Consequently, a quadratic polynomial always has exactly 2 roots.
- The final answer is \(\boxed{2}\).
Explanation
- Identifying the Polynomial Degree
The polynomial in question is given by \(P(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are real numbers and \(a
eq 0\). The degree of a polynomial is determined by the highest power of the variable \(x\), which in this case is 2 because \(a
eq 0\).
- Applying the Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that every non-constant polynomial of degree \(n\) with complex coefficients has exactly \(n\) roots in the complex number system, when each root is counted with its multiplicity. For a quadratic polynomial like \(ax^2 + bx + c\), the degree \(n\) is equal to 2. Therefore, according to the Fundamental Theorem of Algebra, such a polynomial must have exactly 2 roots.
- Conclusion
Since the degree of the polynomial \(ax^2 + bx + c\) is 2, it is guaranteed to have exactly 2 roots, counting any repeated roots as distinct instances of the root. Thus, the missing number in the sentence is 2.
Examples
The Fundamental Theorem of Algebra is fundamental in many areas of mathematics, including engineering and physics. For instance, when designing a control system for a robot, the stability of the system is often determined by the roots of a characteristic polynomial. Knowing that a second-degree (quadratic) polynomial always has exactly two roots helps engineers predict the behavior of the system and ensure it operates correctly without unexpected oscillations or failures. This theorem provides the mathematical assurance that we can always find the necessary information to analyze such systems.