Dividing Polynomials (4x^2 + 13x + 6) By (x + 2) Quotient And Remainder

In the realm of mathematics, polynomial division stands as a fundamental operation, essential for simplifying complex expressions and solving equations. This article delves into the intricacies of dividing the polynomial $(4x^2 + 13x + 6)$ by the binomial $(x + 2)$. We will explore the underlying principles, step-by-step methods, and practical applications of this algebraic maneuver. Whether you're a student grappling with homework assignments or a seasoned mathematician seeking a refresher, this guide aims to provide a comprehensive understanding of polynomial division.

Understanding Polynomial Division

Polynomial division, at its core, is the process of dividing a polynomial by another polynomial of lower or equal degree. It's analogous to long division with numbers, but instead of digits, we're dealing with algebraic terms. The goal is to find the quotient and the remainder, which satisfy the equation:

Dividend = (Divisor × Quotient) + Remainder

In our case, the dividend is $(4x^2 + 13x + 6)$, the divisor is $(x + 2)$, and we aim to find the quotient and remainder.

Methods for Polynomial Division

Several methods exist for dividing polynomials, each with its own advantages and suitability for different scenarios. The most common methods include:

1. Long Division: This method is a systematic, step-by-step approach that mirrors the long division process used for numbers. It's particularly useful for dividing by polynomials of degree 2 or higher.
2. Synthetic Division: This is a shorthand method specifically designed for dividing by linear binomials of the form $(x - a)$. It's more efficient than long division for these cases.

For our problem, $(4x^2 + 13x + 6) ÷ (x + 2)$, both long division and synthetic division can be applied. We will demonstrate both methods for a thorough understanding.

Long Division Method

The long division method is a robust technique for dividing polynomials of any degree. It involves setting up the problem in a similar format to numerical long division and systematically dividing, multiplying, subtracting, and bringing down terms until the quotient and remainder are obtained. Let's apply this method to our problem:

Step-by-Step Process

1. Set up the division: Write the dividend $(4x^2 + 13x + 6)$ inside the division symbol and the divisor $(x + 2)$ outside.

          ________
x + 2 | 4x^2 + 13x + 6

2. Divide the leading terms: Divide the leading term of the dividend $(4x^2)$ by the leading term of the divisor $(x)$. This gives $4x$, which is the first term of the quotient.

          4x ______
x + 2 | 4x^2 + 13x + 6

3. Multiply the quotient term by the divisor: Multiply $4x$ by $(x + 2)$ to get $(4x^2 + 8x)$.

          4x ______
x + 2 | 4x^2 + 13x + 6
       4x^2 + 8x

4. Subtract: Subtract $(4x^2 + 8x)$ from $(4x^2 + 13x)$ to get $5x$.

          4x ______
x + 2 | 4x^2 + 13x + 6
       4x^2 + 8x
       ---------
             5x

5. Bring down the next term: Bring down the next term from the dividend, which is $+6$.

          4x ______
x + 2 | 4x^2 + 13x + 6
       4x^2 + 8x
       ---------
             5x + 6

6. Repeat the process: Divide the new leading term $(5x)$ by the leading term of the divisor $(x)$. This gives $+5$, which is the next term of the quotient.

          4x + 5
x + 2 | 4x^2 + 13x + 6
       4x^2 + 8x
       ---------
             5x + 6

7. Multiply the quotient term by the divisor: Multiply $5$ by $(x + 2)$ to get $(5x + 10)$.

          4x + 5
x + 2 | 4x^2 + 13x + 6
       4x^2 + 8x
       ---------
             5x + 6
             5x + 10

8. Subtract: Subtract $(5x + 10)$ from $(5x + 6)$ to get $-4$.

          4x + 5
x + 2 | 4x^2 + 13x + 6
       4x^2 + 8x
       ---------
             5x + 6
             5x + 10
             ------
                 -4

9. Identify the quotient and remainder: The quotient is $4x + 5$, and the remainder is $-4$.

Result

Therefore, $(4x^2 + 13x + 6) ÷ (x + 2)$ yields a quotient of $4x + 5$ and a remainder of $-4$. This can be expressed as:

$4x^2 + 13x + 6 = (x + 2)(4x + 5) - 4$

Synthetic Division Method

Synthetic division provides a more streamlined approach for dividing polynomials by linear binomials. It simplifies the long division process by focusing on the coefficients of the polynomials. Let's apply synthetic division to our problem:

Step-by-Step Process

1. Identify the divisor's root: The divisor is $(x + 2)$, so the root is $x = -2$. This is the value we'll use in synthetic division.
2. Write down the coefficients: Write down the coefficients of the dividend $(4x^2 + 13x + 6)$, which are $4$, $13$, and $6$.

-2 | 4  13  6
   |
   ----------

3. Bring down the first coefficient: Bring down the first coefficient ($4$) below the line.

-2 | 4  13  6
   |    
   ----------
    4

4. Multiply and add: Multiply the root $(-2)$ by the number just written below the line ($4$) to get $-8$. Write this under the next coefficient ($13$).

-2 | 4  13  6
   |    -8
   ----------
    4

Add $13$ and $-8$ to get $5$, and write this below the line.

-2 | 4  13  6
   |    -8
   ----------
    4   5

5. Repeat the process: Multiply the root $(-2)$ by the new number below the line ($5$) to get $-10$. Write this under the next coefficient ($6$).

-2 | 4  13  6
   |    -8 -10
   ----------
    4   5

Add $6$ and $-10$ to get $-4$, and write this below the line.

-2 | 4  13  6
   |    -8 -10
   ----------
    4   5  -4

6. Identify the quotient and remainder: The last number below the line ($-4$) is the remainder. The other numbers ($4$ and $5$) are the coefficients of the quotient. Since we divided a quadratic by a linear term, the quotient will be linear. Thus, the quotient is $4x + 5$.

Result

Using synthetic division, we again find that $(4x^2 + 13x + 6) ÷ (x + 2)$ yields a quotient of $4x + 5$ and a remainder of $-4$.

Verifying the Result

To ensure our results are correct, we can verify the division using the equation:

Dividend = (Divisor × Quotient) + Remainder

Plugging in our values:

$4x^2 + 13x + 6 = (x + 2)(4x + 5) - 4$

Expanding the right side:

$(x + 2)(4x + 5) - 4 = 4x^2 + 5x + 8x + 10 - 4 = 4x^2 + 13x + 6$

The left side equals the right side, confirming that our division is correct.

Applications of Polynomial Division

Polynomial division is not just an abstract mathematical exercise; it has numerous applications in various fields:

1. Factoring Polynomials: If the remainder of a division is zero, it means the divisor is a factor of the dividend. This is a powerful technique for factoring complex polynomials.
2. Solving Equations: Polynomial division can help simplify equations and find their roots. If we know one root of a polynomial, we can divide the polynomial by the corresponding linear factor to reduce the degree of the polynomial and find other roots.
3. Calculus: Polynomial division is used in calculus for integrating rational functions. By dividing the numerator by the denominator, we can often express a rational function in a form that is easier to integrate.
4. Engineering and Physics: Polynomials are used to model various physical phenomena, and polynomial division can be used to simplify these models and solve related problems.

Conclusion

In conclusion, dividing $(4x^2 + 13x + 6)$ by $(x + 2)$ results in a quotient of $4x + 5$ and a remainder of $-4$. We demonstrated both the long division and synthetic division methods, both of which yielded the same result. Understanding polynomial division is crucial for mastering algebra and its applications in various fields. By mastering the techniques and applications discussed in this guide, you'll be well-equipped to tackle a wide range of mathematical problems involving polynomials. Whether you prefer the systematic approach of long division or the efficiency of synthetic division, the key is to practice and apply these methods to solidify your understanding. This comprehensive guide provides a solid foundation for further exploration of algebraic concepts and their real-world implications.