Synthetic Division Demystified A Step-by-Step Guide

Hey guys! Let's dive into the world of synthetic division and tackle this problem together. Synthetic division is a super handy shortcut for dividing polynomials, especially when you're dividing by a linear expression like (x + 2). It's way less messy than long division, trust me!

Understanding Synthetic Division

So, synthetic division is essentially a streamlined way to divide a polynomial by a linear factor of the form (x - k). It allows us to quickly find the quotient and remainder of the division. Before we jump into the problem, let's break down the steps involved in synthetic division to ensure we fully grasp the mathematical method. First, identify the coefficients of the polynomial and the value of 'k' from the divisor (x - k). Write down the coefficients of the polynomial in a row, and then write 'k' to the left. Bring down the first coefficient, multiply it by 'k', and write the result under the next coefficient. Add these two numbers, and then repeat the process until you reach the last coefficient. The last number is the remainder, and the other numbers are the coefficients of the quotient. Remember, the degree of the quotient is one less than the degree of the original polynomial.

The key to mastering polynomial division with synthetic division lies in understanding the underlying principles. We're essentially using the coefficients of the polynomial and the root of the divisor to perform a series of multiplications and additions. This process cleverly eliminates the variable 'x' and focuses solely on the numerical relationships between the terms. Think of it as a mathematical dance where the coefficients gracefully move and interact to reveal the quotient and remainder. Also, pay close attention to placeholders. If a term is missing in the polynomial (e.g., no x term), you need to include a zero as a placeholder in the synthetic division setup. This ensures that the coefficients are aligned correctly and the division process proceeds smoothly. For example, if you're dividing x^4 + 2x^2 - 1 by x - 1, you would set up the synthetic division with the coefficients 1, 0, 2, 0, and -1. The zeros act as essential placeholders for the missing x^3 and x terms, maintaining the integrity of the calculation.

Synthetic division isn't just a trick; it's a powerful tool rooted in the principles of algebra. It's directly connected to the Factor Theorem and the Remainder Theorem, which are fundamental concepts in polynomial algebra. The Factor Theorem states that if a polynomial f(x) has a factor (x - k), then f(k) = 0. The Remainder Theorem states that when a polynomial f(x) is divided by (x - k), the remainder is f(k). Synthetic division provides a practical way to apply these theorems. The remainder you obtain from synthetic division is exactly the value of the polynomial evaluated at 'k'. If the remainder is zero, then (x - k) is indeed a factor of the polynomial. This connection to the Factor and Remainder Theorems highlights the importance of synthetic division in factoring polynomials and finding their roots. It's a technique that bridges the gap between abstract algebraic concepts and concrete calculations, making it an invaluable tool for students and mathematicians alike.

Let's Solve the Problem

Our problem is to find $(2x^4 + 4x^3 + 2x^2 + 8x + 8) obreakspace ÷ (x + 2)$ using synthetic division. Here’s how we’ll tackle it, step-by-step, making it super easy to follow:

1. Identify the coefficients and the divisor: First, we need to line up our players. The coefficients of our polynomial are 2, 4, 2, 8, and 8. Our divisor is (x + 2), which we can rewrite as (x - (-2)). So, our 'k' value is -2.
2. Set up the synthetic division: Draw a little division symbol (like an upside-down L). Write -2 to the left, and then write the coefficients 2, 4, 2, 8, and 8 across the top row. Make sure you've got enough space to write below each coefficient.
3. Bring down the first coefficient: Bring down the first coefficient (which is 2) below the line. This is our starting point.
4. Multiply and add: Multiply -2 by the number you just brought down (2), which gives you -4. Write -4 under the next coefficient (4). Now, add 4 and -4, which gives you 0. Write 0 below the line.
5. Repeat the process: Multiply -2 by 0 (which is 0), and write it under the next coefficient (2). Add 2 and 0, which gives you 2. Write 2 below the line. Multiply -2 by 2 (which is -4), and write it under the next coefficient (8). Add 8 and -4, which gives you 4. Write 4 below the line. Finally, multiply -2 by 4 (which is -8), and write it under the last coefficient (8). Add 8 and -8, which gives you 0. Write 0 below the line.
6. Interpret the results: The numbers below the line (excluding the last one) are the coefficients of our quotient. The last number is the remainder. So, we have 2, 0, 2, and 4 as our coefficients, and 0 as our remainder. Remember that the quotient will have a degree one less than the original polynomial. Since our original polynomial was degree 4, our quotient will be degree 3.

Let's break down this step-by-step solution to truly understand how synthetic division works. We started by identifying the coefficients of the dividend, which are the numbers in front of the x terms (and the constant term). We also found the root of the divisor by setting x + 2 = 0, giving us x = -2. This -2 is the key number we use in the synthetic division process. The setup is crucial. We arrange the coefficients in a row and place the root (-2) to the left. Then, we bring down the first coefficient, which is 2 in this case. This is where the multiplication and addition magic begins. We multiply the root (-2) by the number we just brought down (2), giving us -4. We write this -4 under the next coefficient (4) and add them together, resulting in 0. This process is repeated for each coefficient. The final row of numbers represents the coefficients of the quotient and the remainder. In this case, the numbers 2, 0, 2, and 4 are the coefficients of the quotient, and the last number, 0, is the remainder. Since the remainder is 0, we know that (x + 2) divides evenly into the polynomial, which means (x + 2) is a factor of the polynomial.

To fully appreciate the power of synthetic division, it's helpful to compare it to long division. Long division can be cumbersome and prone to errors, especially with higher-degree polynomials. Synthetic division, on the other hand, streamlines the process by focusing on the coefficients and the root of the divisor. This makes it faster and less prone to mistakes. The key is to understand the underlying mechanics of the process. Each step has a specific purpose, and by following them carefully, you can efficiently divide polynomials. Remember, practice makes perfect. The more you work through examples, the more comfortable you'll become with synthetic division. You'll start to recognize patterns and understand how the coefficients and the root interact to produce the quotient and remainder. This understanding will not only help you solve problems quickly but also deepen your grasp of polynomial algebra.

The Answer

So, our quotient is $2x^3 + 0x^2 + 2x + 4$, which simplifies to $2x^3 + 2x + 4$. And our remainder is 0. Therefore, $(2x^4 + 4x^3 + 2x^2 + 8x + 8) ÷ (x + 2) = 2x^3 + 2x + 4$.

Thus, the correct answer is A. $2x^3 + 2x + 4$

Key Takeaways

• Synthetic division is a quick and efficient way to divide polynomials by linear expressions.
• Remember to include placeholders (zeros) for missing terms in the polynomial.
• The remainder theorem tells us that the remainder is the value of the polynomial evaluated at the root of the divisor.

Hope this helps you nail synthetic division, guys! Keep practicing, and you'll become a pro in no time!