Real And Imaginary Solutions For The Cubic Equation $x^3 + 3x^2 + 5x + 15 = 0$

When it comes to understanding the nature of solutions for polynomial equations, we often delve into the realms of real and imaginary numbers. The equation $x^2 - 9 = 0$ serves as a simple yet effective illustration of an equation with real solutions. However, when we venture into cubic equations, the landscape becomes more intriguing. In this exploration, we will dissect the equation $x^3 + 3x^2 + 5x + 15 = 0$, aiming to determine the number of real and imaginary solutions it possesses. By employing algebraic techniques and leveraging the fundamental theorem of algebra, we can unravel the mysteries hidden within this cubic equation. This journey will not only enhance our problem-solving skills but also deepen our comprehension of polynomial equations and their multifaceted solutions.

To set the stage for our cubic equation analysis, let's first examine the quadratic equation $x^2 - 9 = 0$. This equation is a classic example of a quadratic, and its solutions can be readily found through various methods, such as factoring or using the quadratic formula. By adding 9 to both sides, we get $x^2 = 9$. Taking the square root of both sides yields $x = ±3$. Thus, we have two distinct real solutions: $x = 3$ and $x = -3$. These solutions represent the points where the parabola defined by $y = x^2 - 9$ intersects the x-axis. The simplicity of this quadratic equation provides a solid foundation for understanding the nature of real solutions. These real solutions are numbers that can be plotted on a number line, making them tangible and intuitive. The graph of the quadratic equation visually confirms these solutions, as the parabola crosses the x-axis at two distinct points. This graphical representation further solidifies our understanding of real solutions and their significance in the context of equations.

Now, let's shift our focus to the cubic equation $x^3 + 3x^2 + 5x + 15 = 0$. Cubic equations, being polynomials of degree three, introduce a new layer of complexity compared to quadratics. The Fundamental Theorem of Algebra plays a pivotal role in understanding the solutions of polynomial equations. This theorem states that a polynomial equation of degree n has exactly n complex roots, counted with multiplicity. This means that our cubic equation, having a degree of 3, must have three solutions, which may be real or imaginary. Imaginary solutions, often arising from taking the square root of negative numbers, expand the solution set beyond the realm of real numbers. These solutions involve the imaginary unit i, where $i^2 = -1$. The interplay between real and imaginary solutions is a defining characteristic of higher-degree polynomial equations. Understanding the Fundamental Theorem of Algebra is crucial for predicting the number and nature of solutions in polynomial equations. It assures us that every polynomial equation has a complete set of solutions within the complex number system.

To determine the number of real and imaginary solutions for the cubic equation $x^3 + 3x^2 + 5x + 15 = 0$, we can employ a technique called factoring by grouping. First, we group the terms in pairs: $(x^3 + 3x^2) + (5x + 15) = 0$. Next, we factor out the greatest common factor from each pair. From the first pair, we can factor out $x^2$, resulting in $x^2(x + 3)$. From the second pair, we can factor out 5, yielding $5(x + 3)$. Now, we have $x^2(x + 3) + 5(x + 3) = 0$. Notice that $(x + 3)$ is a common factor in both terms, so we can factor it out: $(x + 3)(x^2 + 5) = 0$. This factored form allows us to find the solutions by setting each factor equal to zero.

Setting $(x + 3) = 0$, we find the first solution: $x = -3$. This is a real solution. Now, let's consider the second factor, $(x^2 + 5) = 0$. Subtracting 5 from both sides gives us $x^2 = -5$. Taking the square root of both sides yields $x = ±√(-5)$. Since we have the square root of a negative number, these solutions are imaginary. We can rewrite them as $x = ±√5i$, where i is the imaginary unit. Therefore, we have two imaginary solutions: $x = √5i$ and $x = -√5i$. In summary, the cubic equation $x^3 + 3x^2 + 5x + 15 = 0$ has one real solution ($x = -3$) and two imaginary solutions ($x = √5i$ and $x = -√5i$).

Based on our solution process, we've successfully identified the nature and number of solutions for the cubic equation $x^3 + 3x^2 + 5x + 15 = 0$. We found one real solution, $x = -3$, and two imaginary solutions, $x = √5i$ and $x = -√5i$. This outcome aligns perfectly with the Fundamental Theorem of Algebra, which predicts that a cubic equation should have three solutions in total. The presence of imaginary solutions underscores the importance of considering complex numbers when analyzing polynomial equations. Imaginary solutions arise when dealing with the square roots of negative numbers, and they play a crucial role in completing the solution set for many polynomial equations. Understanding the nature and number of solutions provides a comprehensive understanding of the behavior of the equation and its graphical representation. In this specific case, the cubic equation intersects the x-axis at only one point ($x = -3$), which corresponds to the real solution. The imaginary solutions, on the other hand, do not have corresponding x-intercepts on the real number plane. This analysis highlights the interconnectedness between algebraic solutions and their geometric interpretations.

In conclusion, we have thoroughly investigated the equation $x^3 + 3x^2 + 5x + 15 = 0$ and determined that it has 1 real and 2 imaginary solutions. By employing factoring by grouping and leveraging the Fundamental Theorem of Algebra, we were able to systematically find all the solutions. This exploration underscores the significance of understanding both real and imaginary numbers in the context of polynomial equations. The ability to identify and classify solutions as real or imaginary is a crucial skill in advanced algebra and calculus. Furthermore, this analysis provides valuable insights into the behavior of cubic equations and their graphical representations. The single real solution corresponds to the point where the cubic function intersects the x-axis, while the imaginary solutions do not have real-world graphical counterparts. This comprehensive understanding empowers us to tackle more complex polynomial equations and appreciate the rich interplay between algebra and complex numbers. The journey through this cubic equation has not only expanded our problem-solving toolkit but also deepened our appreciation for the elegance and intricacies of mathematical solutions.