Finding The Quotient Of Polynomial Division X^2 + 3x + 2 Divided By X + 1

Jul 10, 2025 by ADMIN 74 views

Understanding Polynomial Division and Quotients

Polynomial division, a cornerstone of algebraic manipulation, allows us to divide one polynomial by another, much like long division with numbers. In this article, we will delve into a specific polynomial division problem, dissecting each step to arrive at the quotient and understand the underlying principles. We will address the question: What is the quotient when $x^2 + 3x + 2$ is divided by $x + 1$ ? This exploration will not only provide the answer but also illuminate the process of polynomial division, a fundamental skill in algebra.

The question is: What is the quotient of the following division problem?

$\begin{array}{r}
 x + 1 \longdiv { x ^ { 2 } + 3 x + 2 } \
 \frac{x^2+x}{0+2 x+2} \
 \frac{2 x+2}{0}
\end{array}$

To effectively address this question, we must first understand the mechanics of polynomial long division and how it mirrors the familiar process of numerical long division. Polynomial division involves dividing a polynomial (the dividend) by another polynomial (the divisor) to obtain a quotient and a remainder. The quotient represents the result of the division, while the remainder is the polynomial left over after the division process is complete. In the provided problem, we are given the dividend, $x^2 + 3x + 2$ , and the divisor, $x + 1$ . Our goal is to determine the quotient.

To begin, it is essential to comprehend the setup of the long division problem. The dividend, $x^2 + 3x + 2$ , is placed inside the division symbol, while the divisor, $x + 1$ , is placed outside. The quotient will be written above the division symbol, aligning terms with their corresponding powers of $x$ . The process involves a series of steps, including dividing, multiplying, subtracting, and bringing down terms, until the degree of the remainder is less than the degree of the divisor. This iterative process is crucial for accurately determining the quotient and understanding the relationship between the dividend, divisor, quotient, and remainder.

The first step in the division process involves dividing the leading term of the dividend ( $x^2$ ) by the leading term of the divisor ( $x$ ). This yields $x$ , which becomes the first term of the quotient. We then multiply the entire divisor ( $x + 1$ ) by this term ( $x$ ), resulting in $x^2 + x$ . This product is then subtracted from the dividend ( $x^2 + 3x + 2$ ). This subtraction is a critical step, as it eliminates the leading term of the dividend and allows us to continue the division process with the remaining terms. The result of this subtraction is $2x + 2$ , which becomes the new dividend for the next iteration.

Next, we repeat the process with the new dividend ( $2x + 2$ ). We divide the leading term of the new dividend ( $2x$ ) by the leading term of the divisor ( $x$ ), which gives us $2$ . This becomes the second term of the quotient. We then multiply the divisor ( $x + 1$ ) by this new term ( $2$ ), resulting in $2x + 2$ . Subtracting this product from the new dividend ( $2x + 2$ ) gives us a remainder of $0$ . This indicates that the division is complete, and the divisor divides evenly into the dividend.

Therefore, the quotient of the division problem is $x + 2$ . This is the polynomial that, when multiplied by the divisor ( $x + 1$ ), yields the dividend ( $x^2 + 3x + 2$ ). The remainder of $0$ confirms that the division is exact, meaning there is no leftover term after the division process. Understanding how to arrive at this quotient is crucial for mastering polynomial division and its applications in algebra.

Step-by-Step Solution

To solve the division problem, we follow the standard polynomial long division procedure. This method allows us to systematically divide the dividend ( $x^2 + 3x + 2$ ) by the divisor ( $x + 1$ ). The step-by-step process ensures that we accurately account for each term and maintain the correct order of operations, leading us to the precise quotient. Each step is critical in simplifying the polynomial and revealing the quotient.

Set up the long division: Write the dividend ( $x^2 + 3x + 2$ ) inside the division symbol and the divisor ( $x + 1$ ) outside. This setup provides a visual structure for the division process, mirroring the long division method used with numbers. Proper alignment of terms is crucial for maintaining clarity and accuracy throughout the process.

x + 1 | x^2 + 3x + 2

Divide the leading terms: Divide the leading term of the dividend ( $x^2$ ) by the leading term of the divisor ( $x$ ). This gives us $x$ , which is the first term of the quotient. This step is fundamental, as it determines the initial term of the quotient and sets the stage for subsequent steps. The result, $x$ , is placed above the division symbol, aligned with the $x$ term of the dividend.

    x
x + 1 | x^2 + 3x + 2

Multiply the divisor by the first term of the quotient: Multiply the entire divisor ( $x + 1$ ) by the first term of the quotient ( $x$ ). This yields $x(x + 1) = x^2 + x$ . This step is crucial for determining the portion of the dividend that is accounted for by the first term of the quotient. The result, $x^2 + x$ , is written below the dividend, aligning like terms.

    x
x + 1 | x^2 + 3x + 2
      x^2 + x

Subtract: Subtract the result ( $x^2 + x$ ) from the corresponding terms of the dividend ( $x^2 + 3x$ ). This gives us $(x^2 + 3x) - (x^2 + x) = 2x$ . This subtraction step is vital for eliminating the leading term of the dividend and revealing the remaining portion to be divided. The result, $2x$ , is brought down as the new leading term.

    x
x + 1 | x^2 + 3x + 2
    - (x^2 + x)
    -----------
          2x

Bring down the next term: Bring down the next term from the dividend ( $+2$ ) to form the new expression $2x + 2$ . This step prepares the remaining terms for further division and ensures that all parts of the dividend are accounted for in the process. The new expression, $2x + 2$ , becomes the focus for the next iteration of the division.

    x
x + 1 | x^2 + 3x + 2
    - (x^2 + x)
    -----------
          2x + 2

Divide the new leading term: Divide the new leading term ( $2x$ ) by the leading term of the divisor ( $x$ ). This gives us $2$ , which is the next term of the quotient. This division step determines the subsequent term of the quotient, building upon the previous steps. The result, $2$ , is added to the quotient above the division symbol.

    x + 2
x + 1 | x^2 + 3x + 2
    - (x^2 + x)
    -----------
          2x + 2

Multiply the divisor by the new term of the quotient: Multiply the divisor ( $x + 1$ ) by the new term of the quotient ( $2$ ). This yields $2(x + 1) = 2x + 2$ . This multiplication step calculates the portion of the new expression that is accounted for by the current term of the quotient. The result, $2x + 2$ , is written below the new expression, aligning like terms.

    x + 2
x + 1 | x^2 + 3x + 2
    - (x^2 + x)
    -----------
          2x + 2
          2x + 2

Subtract: Subtract the result ( $2x + 2$ ) from the current expression ( $2x + 2$ ). This gives us $(2x + 2) - (2x + 2) = 0$ . This final subtraction step demonstrates that the division is complete, as there is no remainder. The result, $0$ , signifies the end of the division process.

    x + 2
x + 1 | x^2 + 3x + 2
    - (x^2 + x)
    -----------
          2x + 2
    - (2x + 2)
    -----------
              0

Identify the quotient: The quotient is the expression above the division symbol, which is $x + 2$ . This is the final result of the polynomial division process. The quotient, $x + 2$ , represents the polynomial that, when multiplied by the divisor, yields the dividend.

Therefore, the quotient of the division problem is $x + 2$ .

Verifying the Solution

To ensure the accuracy of our polynomial division, it is essential to verify the solution. We can do this by multiplying the quotient we obtained ( $x + 2$ ) by the divisor ( $x + 1$ ). If the result of this multiplication equals the dividend ( $x^2 + 3x + 2$ ), then we can confidently conclude that our division was performed correctly. This verification step is a crucial practice in mathematics to confirm the correctness of the solution.

Multiply the quotient ( $x + 2$ ) by the divisor ( $x + 1$ ):

$(x + 2)(x + 1) = x(x + 1) + 2(x + 1)$

$= x^2 + x + 2x + 2$

$= x^2 + 3x + 2$

The result of the multiplication, $x^2 + 3x + 2$ , matches the original dividend. This confirms that our quotient, $x + 2$ , is indeed correct. The verification step not only ensures the accuracy of the solution but also reinforces the relationship between the dividend, divisor, and quotient in polynomial division.

Conclusion

In conclusion, through the process of polynomial long division, we have successfully determined that the quotient of the division problem $(x^2 + 3x + 2) / (x + 1)$ is $x + 2$ . We meticulously followed each step of the long division procedure, ensuring accuracy and clarity. Furthermore, we verified our solution by multiplying the quotient by the divisor, which resulted in the original dividend, thus confirming the correctness of our answer. Understanding polynomial division is a crucial skill in algebra, and this detailed exploration provides a solid foundation for tackling more complex problems in the future. The ability to perform polynomial division accurately and efficiently is essential for various mathematical applications.