Dividing Polynomials A Step-by-Step Guide To (3x^2 - 10x - 8) / (x - 4)

Polynomial division can seem daunting at first, but with a systematic approach and a clear understanding of the underlying principles, it becomes a manageable and even elegant process. In this comprehensive guide, we will delve into the step-by-step process of dividing the polynomial $(3x^2 - 10x - 8)$ by the binomial $(x - 4)$. We'll break down each step, providing clear explanations and illustrative examples to ensure a solid grasp of the concept. By the end of this guide, you'll be equipped with the skills and confidence to tackle similar polynomial division problems with ease.

Understanding Polynomial Division

Before we dive into the specifics of our problem, let's lay the groundwork by understanding the general principles of polynomial division. Polynomial division is analogous to long division with numbers, but instead of dealing with digits, we're working with terms containing variables and exponents. The goal is the same: to find the quotient and the remainder when one polynomial is divided by another.

At its core, polynomial division involves systematically dividing the dividend (the polynomial being divided) by the divisor (the polynomial we're dividing by). The result is the quotient, which represents how many times the divisor goes into the dividend, and the remainder, which is the portion of the dividend that's left over after the division. The dividend, divisor, quotient, and remainder are related by the following equation:

Dividend = (Divisor × Quotient) + Remainder

This equation is crucial for verifying the correctness of our division. Once we've performed the division, we can multiply the divisor by the quotient and add the remainder. The result should equal the original dividend. Now, let’s explore the step-by-step process of polynomial division, applying it specifically to our problem of dividing $(3x^2 - 10x - 8)$ by $(x - 4)$. This will not only clarify the method but also provide a solid foundation for tackling more complex problems in the future. Understanding the mechanics of polynomial division is essential for simplifying expressions, solving equations, and mastering higher-level algebraic concepts.

Step-by-Step Division of $(3x^2 - 10x - 8)$ by $(x - 4)$

Now, let's tackle the problem at hand: dividing $(3x^2 - 10x - 8)$ by $(x - 4)$. We'll follow a step-by-step approach, mirroring the long division process you might be familiar with from arithmetic. Each step will be explained in detail to ensure clarity.

Step 1: Setting Up the Division

First, we set up the division problem in a format similar to long division. Write the dividend $(3x^2 - 10x - 8)$ inside the division symbol and the divisor $(x - 4)$ outside. This visual setup helps organize the terms and keeps track of the division process. Ensuring that both the dividend and divisor are written in descending order of powers of the variable is crucial. This arrangement facilitates the division process by aligning like terms and simplifying the steps involved. In our case, the dividend and the divisor are already in the correct order, making the setup straightforward.

Step 2: Dividing the First Terms

Next, we focus on the first terms of both the dividend and the divisor. Divide the first term of the dividend, $3x^2$, by the first term of the divisor, $x$. This gives us $3x$. Write this term as the first term of the quotient above the division symbol, aligning it with the $x$ term in the dividend. This step is the cornerstone of polynomial division, setting the stage for subsequent operations. By focusing on the leading terms, we systematically reduce the degree of the dividend, bringing us closer to the final quotient and remainder.

Step 3: Multiplying the Quotient Term by the Divisor

Now, multiply the term we just wrote in the quotient, $3x$, by the entire divisor, $(x - 4)$. This gives us $3x(x - 4) = 3x^2 - 12x$. Write this result below the dividend, aligning like terms. This multiplication step is essential for determining the portion of the dividend that is accounted for by the current quotient term. By distributing $3x$ across the divisor, we identify the terms that will be subtracted from the dividend in the next step, thus progressing the division.

Step 4: Subtracting and Bringing Down the Next Term

Subtract the expression we just obtained, $(3x^2 - 12x)$, from the corresponding terms in the dividend, $(3x^2 - 10x)$. This gives us $(3x^2 - 10x) - (3x^2 - 12x) = 2x$. Bring down the next term from the dividend, which is $-8$. This combines with the $2x$ to give us the new expression $2x - 8$. This subtraction step is critical for reducing the dividend and revealing the next portion to be divided. Bringing down the next term ensures that all parts of the dividend are incorporated into the division process, ultimately leading to an accurate quotient and remainder.

Step 5: Repeating the Process

Now, we repeat the process with the new expression, $2x - 8$. Divide the first term, $2x$, by the first term of the divisor, $x$. This gives us $2$. Write this term as the next term in the quotient, after the $3x$. Multiply this term, $2$, by the divisor, $(x - 4)$, which gives us $2(x - 4) = 2x - 8$. Write this below the $2x - 8$ expression. This repetition highlights the iterative nature of polynomial division, where each cycle brings us closer to the final answer. By focusing on the leading terms and systematically reducing the dividend, we methodically uncover the quotient and remainder.

Step 6: Final Subtraction and Remainder

Subtract the expression we just obtained, $(2x - 8)$, from the current expression, $(2x - 8)$. This gives us $(2x - 8) - (2x - 8) = 0$. Since the result is $0$, there is no remainder. This final subtraction reveals whether there is a remainder or if the division is exact. In our case, the result of zero indicates that $(x - 4)$ divides $(3x^2 - 10x - 8)$ evenly, without any remainder.

Step 7: The Quotient

The quotient is the expression we wrote above the division symbol, which is $3x + 2$. Since the remainder is $0$, this means that $(3x^2 - 10x - 8)$ divided by $(x - 4)$ equals $3x + 2$. This is the culmination of our step-by-step process, where the quotient represents the result of the division. A zero remainder confirms that the divisor is a factor of the dividend, a significant observation in algebraic manipulations.

Verifying the Solution

To ensure our solution is correct, we can verify it using the equation:

Dividend = (Divisor × Quotient) + Remainder

In our case, this translates to:

$(3x^2 - 10x - 8) = (x - 4)(3x + 2) + 0$

Expanding the right side, we get:

$(x - 4)(3x + 2) = 3x^2 + 2x - 12x - 8 = 3x^2 - 10x - 8$

This matches our original dividend, confirming that our solution is correct. This verification step is a crucial practice in mathematics, ensuring accuracy and reinforcing understanding. By multiplying the divisor and quotient and adding the remainder, we can confidently confirm the correctness of our polynomial division.

Alternative Methods for Polynomial Division

While the step-by-step method we've outlined is a robust and reliable approach, it's worth noting that other methods can also be used for polynomial division. One such method is synthetic division, which provides a streamlined and often faster way to divide polynomials, especially when the divisor is a linear binomial of the form $(x - a)$.

Synthetic Division

Synthetic division is a shorthand method for dividing polynomials by linear expressions. It is particularly useful when the divisor is of the form $(x - a)$, where $a$ is a constant. Synthetic division simplifies the division process by focusing on the coefficients of the polynomials, rather than the entire terms. This method reduces the amount of writing and can be faster than long division, especially for more complex polynomials. However, it is essential to remember that synthetic division is only applicable when the divisor is a linear binomial, making long division the more versatile method for general polynomial division problems.

To use synthetic division for our problem, we would set it up as follows:

Write down the coefficients of the dividend: 3, -10, -8. Write down the value of $a$ from the divisor $(x - 4)$, which is 4. Perform the synthetic division steps:

Bring down the first coefficient (3). Multiply it by 4 and write the result under the next coefficient (-10). Add these two numbers (-10 and 12) to get 2. Multiply 2 by 4 and write the result under the last coefficient (-8). Add these two numbers (-8 and 8) to get 0.

The resulting numbers (3, 2, 0) represent the coefficients of the quotient and the remainder. In this case, the quotient is $3x + 2$, and the remainder is 0, which matches our previous result.

While synthetic division can be a time-saver, it's important to understand the underlying principles of polynomial division, as the step-by-step method provides a more intuitive understanding of the process.

Conclusion

In this guide, we've explored the process of dividing the polynomial $(3x^2 - 10x - 8)$ by the binomial $(x - 4)$. We've broken down the process into clear, manageable steps, ensuring a thorough understanding of the principles involved. We also verified our solution and touched upon alternative methods like synthetic division. Mastering polynomial division is a crucial step in your mathematical journey, opening doors to more advanced concepts and problem-solving techniques. Whether you prefer the step-by-step method or find synthetic division more efficient, the key is to practice and solidify your understanding. With consistent effort, you'll be able to confidently tackle polynomial division problems and excel in your mathematical endeavors. Remember, the goal is not just to arrive at the correct answer but to understand the underlying logic and principles that make mathematics such a powerful and elegant tool.