Synthetic Division Dividing F(x) = 3x³ - 3x² + 2x + 5 By D(x) = 2x - 1

Hey guys! Today, let's dive into the fascinating world of polynomial division, specifically focusing on synthetic division. We're going to tackle a problem where we have a polynomial f(x) = 3x³ - 3x² + 2x + 5 and we want to divide it by d(x) = 2x - 1. The cool part? We'll be using synthetic division to make our lives easier. So, buckle up and let's get started!

Understanding Polynomial Division and Synthetic Division

Before we jump into the calculations, let's take a moment to understand what polynomial division is all about and how synthetic division simplifies the process. Polynomial division, at its core, is just like the long division you learned back in elementary school, but instead of numbers, we're dealing with polynomials. Think of it as trying to figure out how many times one polynomial fits into another. This process helps us break down complex polynomials into simpler forms, making them easier to analyze and work with.

Now, synthetic division is a streamlined, shortcut method for dividing a polynomial by a linear divisor (something of the form x - a). It's a neat trick that avoids writing out all the variables and exponents, focusing instead on the coefficients. This makes the process faster and less prone to errors. However, it's important to remember that synthetic division only works when dividing by a linear expression. It’s a special and efficient technique, but we need to make sure the conditions are right for its use. The beauty of synthetic division lies in its simplicity and efficiency. By focusing on the numerical coefficients, we can bypass the more cumbersome aspects of traditional polynomial long division. This not only saves time but also reduces the likelihood of making mistakes along the way. For those who find algebraic manipulations daunting, synthetic division is a welcome relief, offering a clear and systematic approach to polynomial division. It transforms a potentially complex task into a series of straightforward arithmetic operations, making it accessible to a wider range of learners and practitioners.

However, it's also important to understand the limitations of synthetic division. While it's incredibly useful for dividing by linear divisors, it cannot be directly applied to divisors of higher degree. In such cases, we must resort to traditional polynomial long division or other more advanced techniques. But for the common scenario of dividing by a linear expression, synthetic division stands out as the method of choice, blending speed, accuracy, and ease of use into one elegant package.

Setting Up Synthetic Division for f(x) = 3x³ - 3x² + 2x + 5 and d(x) = 2x - 1

Okay, let's get practical. Our mission is to divide f(x) = 3x³ - 3x² + 2x + 5 by d(x) = 2x - 1 using synthetic division. The first thing we need to do is find the root of our divisor, d(x). Remember, synthetic division works with the root of the divisor, the value that makes the divisor equal to zero. So, we set 2x - 1 = 0 and solve for x. A quick bit of algebra gives us x = 1/2. This is the magic number we'll be using in our synthetic division setup. This root is the cornerstone of our synthetic division process, acting as the key to unlocking the quotient and remainder of our polynomial division. It’s the value that bridges the gap between the divisor and the dividend, allowing us to perform the division in a streamlined and efficient manner. Finding the root accurately is crucial, as any error here will propagate through the entire process, leading to an incorrect result. Therefore, we must approach this step with care and precision, ensuring that we have the correct value to proceed with.

Next, we need to write down the coefficients of our polynomial, f(x). Make sure you include all the coefficients, even if some terms are missing (in which case, you'd use a zero as a placeholder). In our case, the coefficients are 3, -3, 2, and 5. These coefficients will form the basis of our synthetic division table, guiding us through the arithmetic operations that will reveal the quotient and remainder of the division. They represent the numerical essence of the polynomial, stripped of the variable x and its exponents, allowing us to focus solely on the mathematical relationships between the terms. Ensuring that these coefficients are accurately recorded is paramount to the success of the synthetic division process. Any mistake in transcribing these values will inevitably lead to an incorrect outcome. Therefore, we must double-check our work, verifying that each coefficient is correctly placed and corresponds to the appropriate term in the polynomial. This meticulous attention to detail is a hallmark of sound mathematical practice, ensuring that our efforts yield meaningful and reliable results.

Now, we're ready to set up our synthetic division table. Draw a horizontal line and a vertical line to create a sort of upside-down L shape. Place the root we found (1/2) to the left of the vertical line, and write the coefficients of f(x) (3, -3, 2, 5) to the right of the vertical line, along the top row. We are now primed to begin the calculations that will unravel the mysteries of our polynomial division problem.

Performing the Synthetic Division

Alright, with our setup in place, let's roll up our sleeves and perform the synthetic division! This is where the magic happens, guys. The process involves a series of simple steps: bring down, multiply, add, and repeat.

First, we bring down the first coefficient (3) below the horizontal line. This is our starting point. This initial step sets the stage for the iterative process that follows, initiating the cascade of calculations that will ultimately reveal the quotient and remainder. It’s a simple yet crucial action, grounding the synthetic division process in the numerical reality of the polynomial's coefficients. The value we bring down serves as the seed for the subsequent multiplications and additions, influencing the entire outcome of the division. Therefore, we must execute this step with precision, ensuring that the first coefficient is correctly transcribed and positioned for the calculations ahead.

Next, we multiply the root (1/2) by the number we just brought down (3), which gives us 3/2. We write this result under the second coefficient (-3). This multiplication is the heart of the synthetic division process, weaving the root of the divisor into the fabric of the dividend's coefficients. It’s the step that translates the divisor’s influence onto the polynomial being divided, shaping the quotient and remainder that will emerge. The accuracy of this multiplication is paramount, as any error here will ripple through the remaining calculations, potentially leading to a flawed result. Therefore, we must exercise care and diligence, ensuring that the product is correctly computed and placed in the appropriate position for the next step.

Now, we add the second coefficient (-3) and the result we just wrote down (3/2). This gives us -3/2. We write this sum below the horizontal line. This addition step is where the interaction between the dividend and the divisor truly manifests, shaping the evolving quotient and remainder. It’s the point where the influence of the root is felt, modifying the coefficients of the dividend in a way that reflects the division being performed. The sum we obtain becomes a crucial piece of the puzzle, contributing to the unfolding solution. Therefore, we must perform this addition with precision, ensuring that the signs and values are correctly combined to yield an accurate result. This attention to detail is essential for the integrity of the synthetic division process, safeguarding against errors that could compromise the final answer.

We repeat these steps for the remaining coefficients. Multiply the root (1/2) by the number we just got (-3/2), which gives us -3/4. Write this under the third coefficient (2). Then, add 2 and -3/4, which gives us 5/4. Write this below the line. And finally, multiply the root (1/2) by 5/4, which gives us 5/8. Write this under the last coefficient (5). Add 5 and 5/8, which gives us 45/8. This goes below the line as well. These repeated steps form the rhythmic heart of synthetic division, a cycle of multiplication and addition that systematically unravels the polynomial. Each iteration builds upon the previous one, weaving the root of the divisor into the fabric of the dividend, shaping the evolving quotient and remainder. The accuracy of each step is crucial, as errors can accumulate and distort the final result. Therefore, we must approach this process with focus and care, ensuring that each multiplication and addition is performed with precision. This diligent execution is the key to unlocking the hidden structure of the polynomial division, revealing the quotient and remainder that lie within.

Interpreting the Results

Fantastic! We've completed the synthetic division. Now comes the exciting part – interpreting the results! The numbers below the horizontal line hold the key to our answer. The last number (45/8) is the remainder, and the other numbers (3, -3/2, 5/4) are the coefficients of the quotient. Remember that the quotient will have a degree one less than the original polynomial. Since our original polynomial was a cubic (degree 3), our quotient will be a quadratic (degree 2). The remainder stands alone, a numerical testament to the portion of the dividend that could not be perfectly divided. Together, the quotient and remainder paint a complete picture of the polynomial division, revealing how the dividend breaks down under the influence of the divisor. Therefore, we must carefully decipher these results, ensuring that we correctly identify the coefficients of the quotient and the value of the remainder. This interpretation is the culmination of our efforts, the moment when the hidden structure of the polynomial division is brought to light.

So, our quotient is 3x² - (3/2)x + 5/4, and our remainder is 45/8. But wait! We need to remember that our divisor was 2x - 1, not x - 1/2. This means we need to adjust our quotient. Since we effectively divided by 2 when we used 1/2 as our root, we need to divide the quotient by 2 to compensate. This adjustment is crucial for ensuring the accuracy of our final result, accounting for the subtle difference between the divisor we used in the synthetic division and the original divisor. It’s a reminder that synthetic division, while powerful, requires careful attention to detail and a thorough understanding of the underlying principles. The adjustment is not merely a cosmetic change; it’s a fundamental correction that aligns our answer with the true mathematical relationship between the dividend, divisor, quotient, and remainder. Therefore, we must always remember to check for and perform this adjustment when necessary, ensuring that our final answer reflects the true outcome of the polynomial division.

This gives us a final quotient of (3/2)x² - (3/4)x + 5/8 and a remainder of 45/8. This is the true result of dividing f(x) by d(x). Congratulations, guys! We've successfully navigated the world of synthetic division and found our answer!

Conclusion

Synthetic division is a powerful tool for dividing polynomials, especially when dealing with linear divisors. It simplifies the process and makes it less prone to errors. By understanding the steps involved and remembering to adjust the quotient when necessary, you can confidently tackle polynomial division problems. So, keep practicing, and you'll become a synthetic division pro in no time!