Polynomial Long Division Step-by-Step Solution For (16x³ + 40x² + 72) ÷ (2x + 6)

Polynomial long division might seem daunting at first, but it's a fundamental technique in algebra that allows us to divide polynomials, much like we divide numbers in arithmetic. In this article, we'll delve into the process of using long division to solve the specific problem: (16x³ + 40x² + 72) ÷ (2x + 6). We will break down each step, ensuring clarity and comprehension, and provide additional tips and tricks to master this essential skill. This method is widely used in various mathematical contexts, from simplifying complex expressions to finding roots of polynomial equations, making it a cornerstone of algebraic manipulation. Understanding polynomial long division not only helps in solving specific problems but also enhances your overall algebraic proficiency, enabling you to tackle more advanced concepts with confidence.

Polynomial long division is an extension of the familiar arithmetic long division, adapted for algebraic expressions. The key concept remains the same: we are trying to find how many times one polynomial (the divisor) fits into another polynomial (the dividend). To effectively perform polynomial long division, it's essential to understand the terminology and the steps involved. The dividend is the polynomial being divided (in our case, 16x³ + 40x² + 72), and the divisor is the polynomial we are dividing by (2x + 6). The result of the division is the quotient, and any remaining part is the remainder. The process involves systematically dividing, multiplying, subtracting, and bringing down terms until we reach a point where the degree of the remainder is less than the degree of the divisor. This methodical approach ensures that we account for each term in the polynomial, leading to an accurate quotient and remainder. Mastering this process requires careful attention to detail and a solid understanding of algebraic manipulation, but with practice, it becomes a straightforward and powerful tool in your mathematical arsenal. Moreover, understanding this process is crucial not only for solving mathematical problems but also for building a strong foundation in algebra, which is essential for more advanced mathematical studies.

Let's walk through the long division process for (16x³ + 40x² + 72) ÷ (2x + 6) step by step. First, set up the long division format with the dividend (16x³ + 40x² + 72) inside the division symbol and the divisor (2x + 6) outside. Ensure that the dividend is written in descending order of powers of x, and include any missing terms with a coefficient of 0 (in this case, we have a missing x term, so we can write 16x³ + 40x² + 0x + 72). This ensures that our columns line up correctly during the subtraction steps. Next, divide the leading term of the dividend (16x³) by the leading term of the divisor (2x), which gives 8x². Write this term as the first term of the quotient above the division symbol. Multiply the entire divisor (2x + 6) by 8x², resulting in 16x³ + 48x². Subtract this from the corresponding terms of the dividend. This step eliminates the highest power of x from the dividend, simplifying the expression. The result of the subtraction is -8x². Bring down the next term from the dividend, which is 0x, making the new expression -8x² + 0x. Repeat the process: divide -8x² by 2x, which gives -4x. Write -4x as the next term in the quotient. Multiply the divisor (2x + 6) by -4x, resulting in -8x² - 24x. Subtract this from -8x² + 0x, which gives 24x. Bring down the last term from the dividend, which is 72, making the new expression 24x + 72. Finally, divide 24x by 2x, which gives 12. Write 12 as the last term in the quotient. Multiply the divisor (2x + 6) by 12, resulting in 24x + 72. Subtract this from 24x + 72, which gives 0. Since the remainder is 0, the division is exact. The quotient is 8x² - 4x + 12. This systematic approach ensures that each term is accounted for and the division is performed accurately. With practice, this process becomes more intuitive, allowing for quicker and more efficient polynomial division.

To further clarify the process, let's break down each step with detailed explanations. Step 1: Setup. Arrange the dividend (16x³ + 40x² + 72) and the divisor (2x + 6) in the long division format. It's crucial to include any missing terms with a coefficient of 0 to maintain proper alignment. For example, rewrite the dividend as 16x³ + 40x² + 0x + 72. This step ensures that when we subtract terms, we are comparing like terms, which is essential for accurate calculations. The correct setup is the foundation for a successful long division. Step 2: Divide the Leading Terms. Divide the first term of the dividend (16x³) by the first term of the divisor (2x). This gives 8x². Write 8x² above the division symbol as the first term of the quotient. This step determines the initial term that, when multiplied by the divisor, will eliminate the highest power of x in the dividend. Step 3: Multiply. Multiply the entire divisor (2x + 6) by the term we just found (8x²). This results in 16x³ + 48x². This multiplication distributes the term across the divisor, ensuring that we account for all terms in the divisor. Step 4: Subtract. Subtract the result (16x³ + 48x²) from the corresponding terms of the dividend (16x³ + 40x²). This gives -8x². The subtraction eliminates the leading term of the dividend, reducing the degree of the polynomial we are working with. Step 5: Bring Down. Bring down the next term from the dividend, which is 0x. The new expression becomes -8x² + 0x. Bringing down the next term prepares the dividend for the next iteration of the division process. Step 6: Repeat. Repeat steps 2 through 5 with the new expression (-8x² + 0x). Divide -8x² by 2x to get -4x. Write -4x in the quotient. Multiply (2x + 6) by -4x to get -8x² - 24x. Subtract this from -8x² + 0x to get 24x. Bring down the last term, 72, resulting in 24x + 72. Repeating the process ensures that we continue to reduce the degree of the polynomial until we reach a remainder that is of lower degree than the divisor. Step 7: Final Division. Divide 24x by 2x to get 12. Write 12 in the quotient. Multiply (2x + 6) by 12 to get 24x + 72. Subtract this from 24x + 72 to get 0. Since the remainder is 0, the division is complete. The final step yields the last term of the quotient and confirms whether the division is exact or if there is a remainder.

After performing the long division, we find that the quotient is 8x² - 4x + 12. This means that (16x³ + 40x² + 72) divided by (2x + 6) results in 8x² - 4x + 12 with no remainder. To verify our answer, we can multiply the quotient (8x² - 4x + 12) by the divisor (2x + 6) and check if it equals the dividend (16x³ + 40x² + 72). This verification step is crucial in ensuring the accuracy of our division, as it confirms that the quotient and divisor, when multiplied, reconstruct the original dividend. This method also highlights the inverse relationship between division and multiplication, reinforcing the fundamental principles of algebra. In this case, (8x² - 4x + 12)(2x + 6) = 16x³ + 48x² - 8x² - 24x + 24x + 72 = 16x³ + 40x² + 72, which confirms our solution. Therefore, we can confidently state that 8x² - 4x + 12 is the correct quotient for the given polynomial division problem. This thorough verification process not only validates our solution but also enhances our understanding of polynomial division and its relationship with polynomial multiplication.

While polynomial long division is a straightforward process, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and improve your accuracy. One frequent error is forgetting to include placeholder terms for missing powers of x in the dividend. For instance, in our example (16x³ + 40x² + 72), the x term is missing. Failing to include a 0x term can misalign the columns during subtraction, leading to incorrect calculations. Always ensure that the dividend is written with all powers of x, from the highest to the constant term, using 0 as the coefficient for any missing terms. Another common mistake is errors in the multiplication and subtraction steps. Careless arithmetic mistakes when multiplying the divisor by the quotient term or subtracting the resulting polynomial from the dividend can significantly alter the final answer. It's crucial to double-check each step of the multiplication and subtraction to ensure accuracy. Pay close attention to signs, especially when subtracting negative terms. A third mistake is bringing down the wrong term or forgetting to bring down a term at all. Each term in the dividend must be brought down at the appropriate step in the division process. Skipping a term or bringing down the wrong term can disrupt the entire calculation and lead to an incorrect quotient. Lastly, students often make mistakes when dividing the leading terms. Ensuring that you correctly divide the coefficients and subtract the exponents of x is essential. For example, when dividing 16x³ by 2x, make sure to divide 16 by 2 to get 8 and subtract the exponents (3 - 1) to get x². By being mindful of these common errors and practicing diligently, you can improve your proficiency in polynomial long division and achieve more accurate results.

To truly master polynomial long division, it's helpful to employ certain tips and tricks that can simplify the process and improve accuracy. First, always double-check your setup. Ensure that the dividend and divisor are written correctly, with terms in descending order of powers of x, and that any missing terms are represented with a 0 coefficient. This initial step is crucial for aligning terms correctly throughout the division process. Second, take your time and work neatly. Polynomial long division involves multiple steps, and rushing can lead to careless mistakes. Write each step clearly and organize your work in columns to make it easier to track your calculations. Neatness not only reduces the likelihood of errors but also makes it easier to review your work and identify any mistakes. Third, use the distributive property carefully when multiplying the divisor by the quotient term. Make sure to multiply each term in the divisor by the quotient term and correctly write the result for subtraction. A thorough understanding of the distributive property is fundamental to accurate polynomial multiplication. Fourth, pay close attention to signs during subtraction. Subtracting a negative term is equivalent to adding, and incorrect sign handling is a common source of errors. Use parentheses or write out the subtraction as addition of the negative to help avoid sign mistakes. Fifth, practice regularly. Like any mathematical skill, polynomial long division improves with practice. Work through a variety of problems, starting with simpler ones and gradually moving to more complex ones. Regular practice not only builds proficiency but also enhances your understanding of the underlying concepts. Finally, check your answer by multiplying the quotient by the divisor and adding the remainder (if any). If the result equals the dividend, your division is correct. This verification step is an excellent way to ensure accuracy and build confidence in your problem-solving abilities. By incorporating these tips and tricks into your practice, you can significantly improve your skills in polynomial long division and tackle algebraic problems with greater ease.

In conclusion, mastering polynomial long division is an invaluable skill in algebra. It allows us to divide polynomials efficiently, enabling us to simplify complex expressions, solve equations, and gain a deeper understanding of polynomial behavior. By breaking down the process into manageable steps and practicing regularly, you can become proficient in this technique. In our example, we successfully divided (16x³ + 40x² + 72) by (2x + 6) to find the quotient 8x² - 4x + 12. This methodical approach, involving careful setup, division, multiplication, subtraction, and bringing down terms, is the key to accurate polynomial division. Remember to avoid common mistakes, such as neglecting placeholder terms or making sign errors, and to utilize helpful tips and tricks to streamline the process. The ability to perform polynomial long division not only enhances your problem-solving skills but also provides a solid foundation for more advanced algebraic concepts. It is a fundamental tool in various mathematical applications, from calculus to abstract algebra. Therefore, investing time and effort in mastering polynomial long division will undoubtedly pay off in your mathematical journey. This skill empowers you to tackle complex problems with confidence and precision, making it an essential component of your mathematical toolkit.

The correct answer is D. 8x² - 4x + 12.