Synthetic Division Explained Solve (2x^4 + 4x^3 + 2x^2 + 8x + 8) Divided By (x + 2)

In the realm of polynomial algebra, synthetic division emerges as a streamlined technique for dividing a polynomial by a linear factor of the form $(x - a)$. This method offers a more efficient alternative to long division, particularly when dealing with higher-degree polynomials. In this comprehensive guide, we will embark on a step-by-step journey to divide the polynomial $(2x^4 + 4x^3 + 2x^2 + 8x + 8)$ by the linear factor $(x + 2)$ using synthetic division. This exploration will not only illuminate the practical application of synthetic division but also delve into the underlying principles that make it a powerful tool in polynomial manipulation. Understanding synthetic division is crucial for simplifying complex polynomial expressions, solving polynomial equations, and gaining deeper insights into the behavior of polynomial functions.

Unveiling the Essence of Synthetic Division

At its core, synthetic division serves as a shorthand method for polynomial long division. It leverages the coefficients of the polynomial and the constant term of the linear divisor to expedite the division process. By focusing on the numerical relationships between these values, synthetic division eliminates the need to write out the variables and exponents, thereby reducing the complexity and potential for errors. This method is particularly advantageous when dividing by linear factors, as it provides a clear and organized structure for tracking the quotients and remainders.

The beauty of synthetic division lies in its ability to transform a seemingly intricate division problem into a series of simple arithmetic operations. This transformation not only simplifies the calculations but also provides a visual representation of the division process, making it easier to identify patterns and interpret the results. Synthetic division is an indispensable tool for mathematicians, engineers, and anyone working with polynomial expressions, as it streamlines the process of polynomial division and opens doors to further algebraic manipulations.

Setting the Stage: Preparing for Synthetic Division

Before we plunge into the mechanics of synthetic division, it's crucial to ensure that the polynomial is presented in standard form. This entails arranging the terms in descending order of their exponents, with any missing terms represented by a coefficient of zero. This step is paramount because synthetic division relies on the orderly alignment of coefficients to maintain accuracy.

In our specific case, the polynomial $(2x^4 + 4x^3 + 2x^2 + 8x + 8)$ is already in standard form, exhibiting a complete sequence of terms from the fourth power down to the constant term. This readiness allows us to proceed directly to the next stage of preparing for synthetic division: identifying the divisor's constant term.

The divisor, $(x + 2)$, is a linear expression in the form $(x - a)$. To extract the constant term, we set the divisor equal to zero and solve for $x$. This yields $x = -2$, which is the value we will use in our synthetic division setup. This value, often referred to as the "test value," plays a pivotal role in the synthetic division process, guiding the calculations and ultimately determining the quotient and remainder.

The Synthetic Division Algorithm: A Step-by-Step Guide

Now that we've laid the groundwork, let's delve into the heart of synthetic division – the algorithm itself. This method is elegantly structured, involving a series of steps that systematically reduce the polynomial until we arrive at the quotient and remainder.

1. Setting up the Framework: The first step involves creating a framework to organize the coefficients and calculations. Draw a horizontal line and a vertical line to form an inverted "L" shape. Write the test value (-2 in our case) to the left of the vertical line. Then, write the coefficients of the polynomial (2, 4, 2, 8, 8) along the top row, to the right of the vertical line.

2. Bringing Down the Leading Coefficient: The first coefficient (2) is brought down below the horizontal line. This value will serve as the leading coefficient of the quotient.

3. Multiplying and Adding: Multiply the test value (-2) by the value just brought down (2), which yields -4. Write this result below the second coefficient (4). Add these two numbers (4 and -4) together, resulting in 0. Write this sum below the horizontal line.

4. Repeating the Process: Repeat the multiplication and addition steps for the remaining coefficients. Multiply the test value (-2) by the last sum calculated (0), which yields 0. Write this result below the third coefficient (2). Add these two numbers (2 and 0) together, resulting in 2. Write this sum below the horizontal line.

5. Continuing the Iteration: Continue this process for the remaining coefficients. Multiply the test value (-2) by the last sum calculated (2), which yields -4. Write this result below the fourth coefficient (8). Add these two numbers (8 and -4) together, resulting in 4. Write this sum below the horizontal line.

6. Final Step: Multiply the test value (-2) by the last sum calculated (4), which yields -8. Write this result below the last coefficient (8). Add these two numbers (8 and -8) together, resulting in 0. Write this sum below the horizontal line.

7. Interpreting the Results: The numbers below the horizontal line represent the coefficients of the quotient and the remainder. The last number (0) is the remainder, and the preceding numbers (2, 0, 2, 4) are the coefficients of the quotient. Since we started with a fourth-degree polynomial and divided by a linear factor, the quotient will be a third-degree polynomial. Therefore, the quotient is $2x^3 + 0x^2 + 2x + 4$, which simplifies to $2x^3 + 2x + 4$.

Deciphering the Outcome: Quotient and Remainder

Following the synthetic division algorithm, we arrive at a quotient of $2x^3 + 2x + 4$ and a remainder of 0. This outcome signifies that $(x + 2)$ divides evenly into $(2x^4 + 4x^3 + 2x^2 + 8x + 8)$, leaving no remainder. In other words, $(x + 2)$ is a factor of $(2x^4 + 4x^3 + 2x^2 + 8x + 8)$.

The quotient, $2x^3 + 2x + 4$, represents the result of the division. It is a polynomial of one degree less than the original dividend, reflecting the reduction in degree caused by the division process. The remainder, 0, indicates that the division was exact, with no leftover portion. This complete division is a hallmark of polynomial factorization, where the divisor is a factor of the dividend.

In summary, the synthetic division process has elegantly demonstrated that $(2x^4 + 4x^3 + 2x^2 + 8x + 8)$ divided by $(x + 2)$ yields a quotient of $2x^3 + 2x + 4$ with no remainder. This result not only provides the answer to the division problem but also reveals a fundamental relationship between the polynomials involved.

Selecting the Correct Answer

Based on our meticulous synthetic division, we've determined that $(2x^4 + 4x^3 + 2x^2 + 8x + 8) ext{ divided by } (x + 2)$ results in a quotient of $2x^3 + 2x + 4$ with a remainder of 0. Now, let's examine the answer choices provided and pinpoint the one that aligns with our findings.

• A. $2x^3 + 2x + 4$: This option perfectly matches our calculated quotient. It accurately represents the polynomial obtained after dividing the original expression by $(x + 2)$.
• B. $2x^4 + 2x^2 + 4x$: This option is incorrect. It presents a polynomial of the fourth degree, whereas the quotient should be of the third degree after dividing a fourth-degree polynomial by a linear factor.
• C. 2x^3 + 2x + 4 + rac{1}{x + 2}: This option is also incorrect. It includes a fractional term, suggesting a non-zero remainder. However, our synthetic division clearly showed a remainder of 0.
• D. $2x^3 + 2x^2 + 4$: This option is incorrect as well. It presents a third-degree polynomial, but the coefficients do not match our calculated quotient.

Therefore, the unequivocal correct answer is A. $2x^3 + 2x + 4$. This choice accurately represents the quotient obtained from the synthetic division of $(2x^4 + 4x^3 + 2x^2 + 8x + 8)$ by $(x + 2)$.

Delving Deeper: The Significance of Synthetic Division

Synthetic division extends beyond a mere computational tool; it provides a gateway to deeper insights into polynomial algebra. This method is intricately linked to the Factor Theorem and the Remainder Theorem, two fundamental concepts that illuminate the relationship between polynomial division, roots, and remainders.

The Factor Theorem states that a polynomial $f(x)$ has a factor $(x - a)$ if and only if $f(a) = 0$. In our case, since the remainder is 0 when we divide by $(x + 2)$, the Factor Theorem confirms that $(x + 2)$ is indeed a factor of $(2x^4 + 4x^3 + 2x^2 + 8x + 8)$. This connection allows us to factor polynomials and find their roots more efficiently.

The Remainder Theorem, on the other hand, states that when a polynomial $f(x)$ is divided by $(x - a)$, the remainder is equal to $f(a)$. In our example, if we were to substitute $x = -2$ into the original polynomial, we would find that the result is 0, which is consistent with our calculated remainder. This theorem provides a direct link between polynomial evaluation and the remainder obtained from division.

By mastering synthetic division, one gains not only a powerful computational tool but also a deeper understanding of the underlying principles that govern polynomial behavior. This understanding is invaluable for solving polynomial equations, factoring polynomials, and analyzing polynomial functions.

Concluding Thoughts: Mastering Synthetic Division

In this comprehensive exploration, we've meticulously dissected the process of dividing $(2x^4 + 4x^3 + 2x^2 + 8x + 8)$ by $(x + 2)$ using synthetic division. We've navigated the steps of the algorithm, deciphered the quotient and remainder, and selected the correct answer from the provided options. Moreover, we've delved into the significance of synthetic division, highlighting its connection to the Factor Theorem and the Remainder Theorem.

Synthetic division stands as a testament to the elegance and efficiency of mathematical techniques. It transforms a potentially cumbersome division problem into a streamlined sequence of arithmetic operations. By mastering this method, individuals gain a valuable tool for simplifying polynomial expressions, solving equations, and gaining deeper insights into the world of algebra. This skill is not only essential for academic pursuits but also proves invaluable in various fields that rely on mathematical modeling and analysis.

As you continue your mathematical journey, embrace the power of synthetic division. Practice its application, explore its nuances, and unlock its potential to simplify complex polynomial problems. With consistent effort and a clear understanding of the underlying principles, you'll master this technique and reap the rewards of its efficiency and elegance.