- The Fundamental Theorem of Algebra states that the number of complex roots of a polynomial is equal to its degree \(n\).
- Equation \(y = -6x - 9\) is a degree 1 polynomial, thus it has \(\boxed{1}\) root.
- Equation \(y = 2x^5 - x^4 + 2x^3 - 6x^2 + 2x + 4\) is a degree 5 polynomial, thus it has \(\boxed{5}\) roots.
- Equation \(y = -4x^3 - 10x^2 + 7x + 5\) is a degree 3 polynomial, thus it has \(\boxed{3}\) roots.
Explanation
- Problem Analysis and the Fundamental Theorem of Algebra
We are tasked with determining the number of roots for three specific polynomial equations when they are set to zero (\(y = 0\)). The Fundamental Theorem of Algebra states that a polynomial of degree \(n\) has exactly \(n\) roots in the complex number system, counted with multiplicity. Therefore, for any polynomial of the form \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0\), the total number of complex roots is equal to its degree \(n\).
- Analyzing the Linear Equation
For the first equation, \(y = -6x - 9\), we set it equal to zero: \(-6x - 9 = 0\). This is a linear equation with a degree of \(n = 1\). Solving for \(x\), we get \(x = -\frac{9}{6} = -1.5\). Since the degree is 1, it has exactly 1 root.
- Analyzing the Quintic Equation
The second equation is \(y = 2x^5 - x^4 + 2x^3 - 6x^2 + 2x + 4\). This is a quintic (fifth-degree) polynomial, meaning \(n = 5\). According to the Fundamental Theorem of Algebra, this polynomial will have 5 roots in the complex number system. Being an odd-degree polynomial, it is guaranteed to have at least one real root.
- Analyzing the Cubic Equation
The third equation is \(y = -4x^3 - 10x^2 + 7x + 5\). This is a cubic (third-degree) polynomial, meaning \(n = 3\). According to the Fundamental Theorem of Algebra, this polynomial will have 3 roots in the complex number system. Similar to the quintic equation, being an odd-degree polynomial, it must have at least one real root.
- Conclusion
Based on the degree of each polynomial, we conclude the number of complex solutions for each:
- For \(y = -6x - 9\), the number of roots is \(\boxed{1}\).
- For \(y = 2x^5 - x^4 + 2x^3 - 6x^2 + 2x + 4\), the number of roots is \(\boxed{5}\).
- For \(y = -4x^3 - 10x^2 + 7x + 5\), the number of roots is \(\boxed{3}\).
Examples
Polynomials are fundamental tools in modeling physical phenomena. For example, a quadratic polynomial might describe the trajectory of a ball thrown into the air, where the roots represent the times at which the ball is at ground level. Cubic or higher-degree polynomials can model more complex systems, such as the stress distribution in a bridge or the fluctuation of populations in an ecosystem. By determining the number of roots, engineers and scientists can predict how many equilibrium states or critical events exist within a system.