Finding The Roots Of F(x) = (x² + 2x - 15)(x² + 8x + 17)

In the realm of mathematics, particularly in algebra, finding the roots of a polynomial function is a fundamental concept. The roots, also known as zeros, are the values of 'x' for which the function f(x) equals zero. This article delves into the process of identifying these roots for the polynomial function f(x) = (x² + 2x - 15)(x² + 8x + 17). We will systematically explore each factor of the polynomial, applying various algebraic techniques to arrive at the complete set of roots. This exploration will not only provide the solution to this specific problem but also offer a broader understanding of how to tackle similar challenges in polynomial algebra. By understanding the methods used, readers can confidently approach complex polynomial equations and extract the valuable information they hold. Let's begin our journey into the fascinating world of polynomial roots.

Deconstructing the Polynomial Function

The polynomial function given is f(x) = (x² + 2x - 15)(x² + 8x + 17). To find the roots, we need to determine the values of x that make f(x) = 0. This can be achieved by setting each factor of the polynomial equal to zero and solving for x. The function is composed of two quadratic factors: (x² + 2x - 15) and (x² + 8x + 17). Each of these quadratic expressions can potentially yield two roots, either real or complex. Therefore, the complete set of roots for the given polynomial will consist of the roots obtained from each quadratic factor. This approach simplifies the problem by breaking it down into smaller, more manageable parts. Understanding this decomposition is crucial in solving polynomial equations efficiently. The process involves applying techniques such as factoring, the quadratic formula, and complex number manipulation. By mastering these techniques, one can confidently tackle various polynomial root-finding problems.

Solving the First Quadratic Factor: x² + 2x - 15

Our first task is to solve the quadratic equation x² + 2x - 15 = 0. This can be achieved by factoring the quadratic expression. We are looking for two numbers that multiply to -15 and add up to 2. These numbers are 5 and -3. Therefore, the quadratic expression can be factored as (x + 5)(x - 3). Setting each factor equal to zero, we get x + 5 = 0 and x - 3 = 0. Solving these linear equations, we find the roots to be x = -5 and x = 3. These are two real roots of the polynomial function. Factoring is a powerful technique for solving quadratic equations, especially when the roots are integers or simple fractions. It allows us to break down the quadratic into two linear factors, making it easier to isolate the variable x. However, not all quadratic equations can be easily factored. In such cases, we turn to the quadratic formula, a versatile tool that provides the roots for any quadratic equation.

Tackling the Second Quadratic Factor: x² + 8x + 17

Now, let's move on to the second quadratic factor: x² + 8x + 17. Setting this equal to zero, we have x² + 8x + 17 = 0. This quadratic expression does not factor easily, so we will employ the quadratic formula to find its roots. The quadratic formula is given by x = (-b ± √(b² - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. In this case, a = 1, b = 8, and c = 17. Substituting these values into the quadratic formula, we get:

x = (-8 ± √(8² - 4 * 1 * 17)) / (2 * 1)

x = (-8 ± √(64 - 68)) / 2

x = (-8 ± √(-4)) / 2

Since the discriminant (b² - 4ac) is negative, the roots will be complex. We can simplify √(-4) as 2i, where 'i' is the imaginary unit (√-1). Therefore, we have:

x = (-8 ± 2i) / 2

x = -4 ± i

This gives us two complex roots: x = -4 + i and x = -4 - i. The quadratic formula is a fundamental tool in algebra, allowing us to find the roots of any quadratic equation, regardless of whether they are real or complex. When the discriminant is negative, it indicates that the roots are complex conjugates, meaning they have the form a + bi and a - bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit.

Assembling the Complete List of Roots

Having solved both quadratic factors, we can now assemble the complete list of roots for the polynomial function f(x) = (x² + 2x - 15)(x² + 8x + 17). From the first factor (x² + 2x - 15), we obtained the real roots x = -5 and x = 3. From the second factor (x² + 8x + 17), we found the complex roots x = -4 + i and x = -4 - i. Therefore, the complete list of roots for the polynomial function is -5, 3, -4 + i, and -4 - i. This set of roots fully characterizes the solutions to the equation f(x) = 0. Understanding the nature of roots, whether real or complex, is crucial in many areas of mathematics and its applications. Real roots correspond to points where the graph of the polynomial intersects the x-axis, while complex roots do not have a direct graphical interpretation in the real plane. However, complex roots are essential in understanding the complete behavior of the polynomial function.

Conclusion: Mastering Polynomial Root Finding

In conclusion, we have successfully identified the complete list of roots for the polynomial function f(x) = (x² + 2x - 15)(x² + 8x + 17). The roots are -5, 3, -4 + i, and -4 - i. This process involved factoring the first quadratic expression, applying the quadratic formula to the second, and understanding the nature of complex roots. The techniques employed in this article are applicable to a wide range of polynomial equations. Mastering these techniques is essential for anyone studying algebra and related fields. The ability to find the roots of a polynomial is not only a fundamental skill in mathematics but also has applications in various fields such as engineering, physics, and computer science. By understanding the underlying principles and practicing these techniques, one can confidently tackle complex polynomial problems and unlock their hidden solutions. The journey through polynomial root finding is a rewarding one, offering insights into the intricate world of algebraic functions and their applications.