Polynomial Long Division: Solve (x^2 - 6x - 16) / (x + 2)

Polynomial long division might seem daunting at first, but it's a straightforward process once you grasp the steps. This article breaks down how to solve the problem $(x^2 - 6x - 16) / (x + 2)$ using polynomial long division. We'll walk through each step in detail, making it easy to follow along, whether you're a student tackling algebra or just brushing up on your math skills. By the end, you'll not only understand the solution but also the underlying method, empowering you to tackle similar problems with confidence. So, let's dive in and demystify polynomial long division together!

Understanding Polynomial Long Division

Before diving into the specific problem, let’s understand what polynomial long division is. Polynomial long division is a method for dividing polynomials, similar to how you divide numbers using long division. It's particularly useful when you need to divide by a polynomial of degree one or higher. The goal is to find the quotient and the remainder when one polynomial is divided by another. In essence, polynomial long division helps us break down complex polynomial fractions into simpler terms. It’s a crucial skill in algebra and calculus, often used for factoring polynomials, finding roots, and simplifying rational expressions. The process involves several steps, including setting up the division, dividing the leading terms, multiplying, subtracting, and bringing down the next term. Each step is methodical and builds upon the previous one, making it a reliable method for solving polynomial division problems.

To better grasp this method, think of it like traditional long division but with variables and exponents involved. Just as with numerical long division, we focus on dividing term by term, working our way from the highest degree to the lowest. This systematic approach ensures that we account for all terms in the polynomial and arrive at the correct quotient and remainder. Understanding this fundamental concept will make the step-by-step process we're about to explore much clearer and more intuitive.

Setting Up the Problem

Now, let's set up our specific problem: $(x^2 - 6x - 16) / (x + 2)$. The first step in polynomial long division is to correctly set up the problem. Think of it like setting up a traditional long division problem with numbers. You'll have the dividend (the polynomial being divided) inside the division symbol and the divisor (the polynomial we're dividing by) outside. For our problem, the dividend is $x^2 - 6x - 16$, and the divisor is $x + 2$. So, we write it as:

        ________
x + 2 | x^2 - 6x - 16

This setup is crucial because it organizes our work and helps us keep track of each step. It visually represents the division process, making it easier to follow along and avoid mistakes. Ensure that both the dividend and divisor are written in descending order of their exponents. This means starting with the highest power of x and moving down to the constant term. In our case, $x^2 - 6x - 16$ is already in the correct order. Similarly, $x + 2$ is also correctly ordered. Proper setup ensures that we divide the correct terms and maintain the correct alignment throughout the process. Taking the time to set up the problem accurately will save you from potential errors later on.

Step-by-Step Solution

Let's go through the step-by-step solution of the polynomial long division for $(x^2 - 6x - 16) / (x + 2)$. This is where the actual division process happens, and each step builds upon the previous one. Follow along carefully to understand how each term is derived.

Step 1: Divide the Leading Terms

First, we focus on the leading terms of both the dividend ($x^2$) and the divisor ($x$). Divide the leading term of the dividend by the leading term of the divisor: $x^2 / x = x$. This result, $x$, becomes the first term of our quotient. Write this $x$ above the $-6x$ term in the division setup:

        x______
x + 2 | x^2 - 6x - 16

This step is the foundation of the entire process. By focusing solely on the leading terms, we simplify the division and break it down into manageable parts. It’s essential to perform this division accurately, as it sets the stage for the subsequent steps. Double-checking this initial division can prevent errors from propagating through the rest of the solution.

Step 2: Multiply the Quotient Term by the Divisor

Next, multiply the $x$ (the first term of our quotient) by the entire divisor $(x + 2)$: $x * (x + 2) = x^2 + 2x$. This result, $x^2 + 2x$, is what we'll subtract from the dividend in the next step. Write this result below the corresponding terms of the dividend:

        x______
x + 2 | x^2 - 6x - 16
        x^2 + 2x

Multiplying the quotient term by the divisor helps us determine what portion of the dividend we’ve accounted for. It’s a crucial step in narrowing down the remainder. Ensuring accuracy in this multiplication is key to the success of the long division process. Take your time to distribute the term correctly and double-check your work to avoid errors that could affect the final answer.

Step 3: Subtract and Bring Down

Now, subtract the result ($x^2 + 2x$) from the corresponding terms of the dividend ($x^2 - 6x$). Remember to distribute the negative sign: $(x^2 - 6x) - (x^2 + 2x) = x^2 - 6x - x^2 - 2x = -8x$. Write this result below. Then, bring down the next term from the dividend, which is -16:

        x______
x + 2 | x^2 - 6x - 16
        x^2 + 2x
        -------
            -8x - 16

Subtraction is a critical step as it determines the new dividend for the next iteration of the division process. Pay close attention to signs during this step to avoid common mistakes. Bringing down the next term ensures that we continue to account for all parts of the original dividend. This combination of subtraction and bringing down creates a new expression that we can divide again, allowing us to continue working towards the final quotient and remainder.

Step 4: Repeat the Process

Now, repeat the process using the new expression, $-8x - 16$. Divide the leading term $-8x$ by the leading term of the divisor $x$: $-8x / x = -8$. This result, -8, is the next term of our quotient. Write this -8 next to the $x$ in the quotient:

        x - 8__
x + 2 | x^2 - 6x - 16
        x^2 + 2x
        -------
            -8x - 16

Continuing the process iteratively allows us to systematically break down the polynomial division problem. Each iteration brings us closer to the final answer by reducing the degree of the remaining dividend. This repetitive nature of polynomial long division makes it a methodical and reliable technique, provided each step is executed with care and precision.

Step 5: Multiply and Subtract Again

Multiply the new quotient term -8 by the divisor $(x + 2)$: $-8 * (x + 2) = -8x - 16$. Write this below the current expression $-8x - 16$:

        x - 8__
x + 2 | x^2 - 6x - 16
        x^2 + 2x
        -------
            -8x - 16
            -8x - 16

Subtract this result from the current expression: $(-8x - 16) - (-8x - 16) = -8x - 16 + 8x + 16 = 0$. This gives us a remainder of 0:

        x - 8__
x + 2 | x^2 - 6x - 16
        x^2 + 2x
        -------
            -8x - 16
            -8x - 16
            -------
                 0

When we reach a remainder of 0, it signifies that the division is exact, and we have successfully found the quotient without any remainder. This is a clean and satisfying result, indicating that the divisor divides evenly into the dividend. A remainder of 0 simplifies further calculations and confirms the accuracy of our steps.

The Result

Since the remainder is 0, the result of the division is simply the quotient we found: $x - 8$. Therefore, $(x^2 - 6x - 16) / (x + 2) = x - 8$. This means that the polynomial $x^2 - 6x - 16$ can be factored into $(x + 2)(x - 8)$.

Understanding the result in terms of factorization can be highly valuable. It allows us to rewrite the original polynomial in a factored form, which can be useful for solving equations, simplifying expressions, and understanding the roots of the polynomial. In this case, recognizing that $x^2 - 6x - 16$ factors into $(x + 2)(x - 8)$ provides additional insight into the behavior and properties of the polynomial.

Conclusion

In conclusion, polynomial long division is a methodical process that allows us to divide polynomials just like we divide numbers. By following the steps of dividing the leading terms, multiplying, subtracting, and bringing down, we can find the quotient and remainder. In the case of $(x^2 - 6x - 16) / (x + 2)$, the result is $x - 8$ with no remainder. This means that $x - 8$ is the quotient, and the division is exact. Understanding and mastering polynomial long division is a valuable skill in algebra and calculus, enabling you to solve a wide range of problems involving polynomial expressions.

Polynomial long division is more than just a mechanical process; it's a fundamental technique that enhances our understanding of polynomial relationships. It provides a structured way to simplify complex expressions, identify factors, and solve equations. Whether you're a student learning the basics or a professional applying advanced mathematical concepts, proficiency in polynomial long division is an asset that will serve you well.

For further learning and practice, you might find helpful resources on websites like Khan Academy's Algebra Section, where you can find more examples and exercises on polynomial division.