Polynomial Division Step By Step Solve (6x^2 + 5x + 1) ÷ (x + 2)

Polynomial division can seem daunting at first, but with a systematic approach, it becomes a manageable task. In this comprehensive guide, we will walk through the process of solving the polynomial division problem: $(6x^2 + 5x + 1) ÷ (x + 2)$. This detailed explanation will not only provide the solution but also help you understand the underlying principles, making similar problems easier to tackle in the future. Whether you're a student grappling with algebra or someone looking to refresh their mathematical skills, this guide will offer clear, step-by-step instructions to master polynomial division.

The problem we aim to solve is to divide the quadratic polynomial $6x^2 + 5x + 1$ by the linear polynomial $x + 2$. The goal is to find the quotient and any remainder that may result from this division. The process we'll use is similar to long division with numbers, but instead of digits, we're working with terms involving the variable $x$. This method ensures that we account for each term in the polynomial and correctly distribute the division. By understanding this process, you’ll be able to confidently approach other polynomial division problems, regardless of their complexity. We will cover each step in detail, from setting up the division to interpreting the final result. So, let's dive in and break down the solution together.

Step 1: Setting Up the Long Division

The first step in solving any polynomial division problem is to set up the long division. This setup mirrors the long division you may have learned with numbers, but with polynomials instead. To begin, write the dividend ($6x^2 + 5x + 1$) inside the division symbol and the divisor ($x + 2$) outside. Ensure that both the dividend and the divisor are written in descending order of powers of $x$. This organization is crucial for keeping track of terms and performing the division systematically. The dividend is the polynomial being divided, and the divisor is the polynomial we are dividing by. Setting it up correctly at the beginning is paramount to avoiding errors later in the process. It's a simple step, but it lays the foundation for the rest of the solution. Once you've set up the division, you're ready to start the iterative process of dividing, multiplying, and subtracting, which will eventually lead you to the quotient and remainder. Remember, accuracy in this initial step can save you from unnecessary complications further on.

To set up the long division, draw the long division symbol, place the dividend $6x^2 + 5x + 1$ inside, and the divisor $x + 2$ outside to the left.

 x + 2 | 6x^2 + 5x + 1

This setup visually organizes the problem and prepares us for the next steps in the division process.

Step 2: Dividing the Leading Terms

Now that we have the setup in place, the next step involves dividing the leading term of the dividend by the leading term of the divisor. In our problem, the leading term of the dividend is $6x^2$, and the leading term of the divisor is $x$. To perform this division, we ask ourselves: "What do we need to multiply $x$ by to get $6x^2$?" The answer is $6x$. This term, $6x$, becomes the first term of our quotient, which we write above the division symbol, aligned with the $x$ term of the dividend. This step is crucial because it starts the process of breaking down the dividend into manageable parts. By focusing on the leading terms, we systematically reduce the complexity of the polynomial. The quotient we obtain in this step will be progressively built upon as we continue the division process. Remember, the key is to focus on the terms with the highest powers of $x$ to ensure an organized and accurate solution.

Divide the leading term of the dividend ($6x^2$) by the leading term of the divisor ($x$).

$6x^2 ÷ x = 6x$.

Write $6x$ above the division symbol, aligned with the $x$ term in the dividend.

 6x
 x + 2 | 6x^2 + 5x + 1

Step 3: Multiplying the Quotient Term by the Divisor

Once we've determined the first term of the quotient, which in our case is $6x$, the next step is to multiply this term by the entire divisor. This step is vital for understanding how much of the dividend we've accounted for and what remains to be divided. We multiply $6x$ by $(x + 2)$, distributing $6x$ across both terms of the divisor. This gives us $6x * x + 6x * 2$, which simplifies to $6x^2 + 12x$. The result, $6x^2 + 12x$, is then written below the dividend, aligning like terms. This alignment is crucial for the subsequent subtraction step, as it ensures we're combining terms with the same powers of $x$. Multiplying the quotient term by the divisor allows us to create a new polynomial that we can subtract from the dividend, thus reducing the complexity of the problem step by step. Accurate multiplication here is key to a correct final answer. This process of multiplying the quotient term by the divisor is a cornerstone of polynomial long division, and mastering it is essential for solving more complex problems.

Multiply the quotient term ($6x$) by the divisor ($x + 2$).

$6x * (x + 2) = 6x^2 + 12x$.

Write the result below the dividend, aligning like terms.

 6x
 x + 2 | 6x^2 + 5x + 1
 6x^2 + 12x

Step 4: Subtracting and Bringing Down the Next Term

Following the multiplication, the next critical step is subtraction. We subtract the polynomial we just obtained ($6x^2 + 12x$) from the corresponding terms in the dividend ($6x^2 + 5x + 1$). This subtraction is performed term by term, ensuring that like terms are combined. Subtracting $6x^2 + 12x$ from $6x^2 + 5x$ gives us $(6x^2 - 6x^2) + (5x - 12x)$, which simplifies to $-7x$. The $6x^2$ terms cancel out, as they should, leaving us with a new term involving $x$. After performing the subtraction, we bring down the next term from the original dividend, which in this case is $+1$. This process of subtraction and bringing down the next term is essential for iteratively reducing the dividend until we either reach a constant term or a remainder that cannot be further divided by the divisor. Subtracting accurately and bringing down the correct term sets the stage for the next cycle of division, ensuring we continue towards the correct solution. This step is a cornerstone of the long division process and requires careful attention to detail to avoid errors.

Subtract the result from the corresponding terms in the dividend.

$(6x^2 + 5x) - (6x^2 + 12x) = -7x$.

Bring down the next term (+1) from the dividend.

 6x
 x + 2 | 6x^2 + 5x + 1
 6x^2 + 12x
 ---------
 -7x + 1

Step 5: Repeating the Process

With our new expression, $-7x + 1$, we repeat the process we followed in steps two through four. This iterative process is the heart of polynomial long division, allowing us to systematically reduce the complexity of the problem until we arrive at the quotient and remainder. First, we divide the leading term of the new expression, which is $-7x$, by the leading term of the divisor, which is $x$. The result of this division, $-7x ÷ x$, is $-7$. This becomes the next term in our quotient, which we add to the $6x$ we found earlier, placing it above the constant term of the dividend. Next, we multiply this new quotient term, $-7$, by the entire divisor, $(x + 2)$. This gives us $-7 * (x + 2)$, which equals $-7x - 14$. We write this result below our current expression, $-7x + 1$, aligning like terms. The last part of this cycle is to subtract the result from the current expression. Subtracting $(-7x - 14)$ from $(-7x + 1)$ yields $(-7x - (-7x)) + (1 - (-14))$, which simplifies to $15$. The $-7x$ terms cancel out, and we are left with a constant, $15$. Repeating this process allows us to systematically break down the polynomial division problem into smaller, manageable steps, ultimately leading to the solution. By carefully following each iteration, we can confidently handle even complex polynomial divisions.

Repeat the process by dividing the leading term of the new expression (-7x) by the leading term of the divisor ($x$).

$-7x ÷ x = -7$.

Add $-7$ to the quotient above the division symbol.

 6x - 7
 x + 2 | 6x^2 + 5x + 1
 6x^2 + 12x
 ---------
 -7x + 1

Multiply the new quotient term (-7) by the divisor ($x + 2$).

$-7 * (x + 2) = -7x - 14$.

Write the result below the current expression, aligning like terms.

 6x - 7
 x + 2 | 6x^2 + 5x + 1
 6x^2 + 12x
 ---------
 -7x + 1
 -7x - 14

Subtract the result from the current expression.

$(-7x + 1) - (-7x - 14) = 15$.

Step 6: Determining the Quotient and Remainder

After completing the iterative process of division, multiplication, and subtraction, we arrive at the final step: determining the quotient and the remainder. In our problem, after subtracting $-7x - 14$ from $-7x + 1$, we were left with $15$. This constant term, $15$, represents the remainder of the division. The remainder is the part of the dividend that could not be evenly divided by the divisor. The quotient, on the other hand, is the polynomial we obtained above the division symbol throughout our steps. In our case, the quotient is $6x - 7$. Now, we express the final result by writing the quotient followed by the remainder as a fraction over the divisor. This is a standard way of representing the outcome of polynomial division, clearly showing both the polynomial that resulted from the division and the portion that remained undivided. Understanding how to identify and express the quotient and remainder is crucial for fully grasping the result of polynomial division. This step completes the solution, providing a comprehensive answer to the original problem.

Since the degree of the remainder (15) is less than the degree of the divisor ($x + 2$), we stop the division.

The quotient is $6x - 7$, and the remainder is $15$.

Step 7: Expressing the Final Result

Now that we've determined the quotient and remainder, the final step is to express the result in a standard format. This involves writing the quotient we obtained ($6x - 7$) and adding the remainder as a fraction over the divisor. The remainder, which is $15$, becomes the numerator of the fraction, and the divisor, $(x + 2)$, becomes the denominator. The complete expression is then written as 6x - 7 + rac{15}{x + 2}. This format clearly shows the result of the polynomial division, with the quotient representing the whole part of the division and the fractional part representing the remainder. Expressing the result in this manner provides a concise and understandable answer to the original division problem. It highlights the relationship between the dividend, divisor, quotient, and remainder, offering a comprehensive solution. This final step consolidates our work, ensuring that the answer is presented clearly and accurately.

Express the final result as the quotient plus the remainder over the divisor.

Final Result:

6x - 7 + rac{15}{x + 2}.

Thus, (6x^2 + 5x + 1) ÷ (x + 2) = 6x - 7 + rac{15}{x + 2}.

Conclusion

In this detailed guide, we've walked through the step-by-step process of solving the polynomial division problem $(6x^2 + 5x + 1) ÷ (x + 2)$. From setting up the long division to determining the quotient and remainder, each step was carefully explained to provide a clear understanding of the method. We began by organizing the problem, then systematically divided, multiplied, subtracted, and brought down terms until we reached a final result. The key takeaways from this guide are the methodical approach to long division, the importance of aligning like terms, and the iterative nature of the process. Understanding these principles allows for solving a wide range of polynomial division problems with confidence. The final result, 6x - 7 + rac{15}{x + 2}, represents the quotient and the remainder in a clear and concise format. By following this guide, you can master polynomial division and tackle similar problems effectively. Practice and familiarity with these steps will enhance your algebraic skills and problem-solving abilities. Remember, polynomial division, like any mathematical concept, becomes easier with understanding and repetition. So, continue to apply these techniques to various problems to solidify your knowledge and build your confidence in algebra.