- Identify the first root \(x=1\) by testing potential rational divisors of the constant term.
- Factor the polynomial as \((x-1)(x^2 + 6x + 9) = 0\) using polynomial division.
- Solve the quadratic factor \((x+3)^2 = 0\) to find the root \(x=-3\).
- Conclude that there are two distinct real solutions: \(\boxed{2}\).
Explanation
- Identifying the First Root
To determine the number of real solutions for the equation \(x^3 + 5x^2 + 3x - 9 = 0\), we first identify potential rational roots using the Rational Root Theorem. The possible rational roots are divisors of the constant term \(-9\), which are \(\pm 1, \pm 3, \pm 9\). Testing \(x=1\), we find \(1^3 + 5(1)^2 + 3(1) - 9 = 1 + 5 + 3 - 9 = 0\), which confirms that \(x=1\) is a root of the equation.
- Factoring the Polynomial
Since \(x=1\) is a root, we know that \((x-1)\) must be a factor of the polynomial \(x^3 + 5x^2 + 3x - 9\). By performing polynomial division, we can express the cubic equation as:
\[(x-1)(x^2 + 6x + 9) = 0\]
This simplifies the cubic equation into a product of a linear factor and a quadratic factor.
- Finding Remaining Roots
Next, we solve the quadratic factor \(x^2 + 6x + 9 = 0\). This quadratic is a perfect square trinomial, which can be factored as \((x+3)^2 = 0\). Thus, \(x = -3\) is the remaining root, with a multiplicity of 2. Combining this with the root \(x=1\), we have identified the complete set of roots for the given equation.
- Conclusion
The equation \(x^3 + 5x^2 + 3x - 9 = 0\) is equivalent to \((x-1)(x+3)^2 = 0\). The distinct real solutions to the equation are \(x = 1\) and \(x = -3\). Therefore, the equation has two distinct real number solutions.
Examples
In physics, when modeling the trajectory of a projectile or the energy levels of a system, you often encounter polynomial equations. For example, if the height of a jumping robot at time \(t\) is modeled by a function \(h(t) = -t^3 + 5t^2 + 3t - 9\), solving \(h(t) = 0\) helps identify the exact times when the robot is at ground level. This analysis allows engineers to predict when a device will touch a surface, ensuring precise control and interaction with the environment.