Polynomial Division Finding The Quotient Of (2x³ - 2x - 12) By (x - 2)
In the realm of algebra, polynomial division stands as a fundamental operation, particularly when dealing with expressions involving variables raised to various powers. This article delves into the process of dividing one polynomial by another, specifically focusing on the quotient obtained when dividing (2x³ - 2x - 12) by (x - 2). Polynomial division, akin to long division with numbers, unveils the underlying structure of polynomials and their relationships. Mastering this technique is crucial for simplifying expressions, solving equations, and gaining a deeper understanding of algebraic concepts.
Understanding Polynomial Division
Before we dive into the specific problem, it's essential to grasp the core principles of polynomial division. Think of it as the reverse process of polynomial multiplication. When we multiply two polynomials, we distribute each term of one polynomial across all terms of the other. Division, on the other hand, aims to undo this process, allowing us to identify the factors that, when multiplied, result in the original polynomial. Polynomial division is a critical skill in algebra, enabling us to simplify complex expressions and solve equations more effectively. In essence, it involves dividing a polynomial (the dividend) by another polynomial (the divisor) to obtain a quotient and a remainder. The quotient represents the result of the division, while the remainder is the portion that's "left over" when the division isn't exact. This process is particularly useful for factoring polynomials, finding roots, and simplifying rational expressions.
The process typically involves these key steps:
- Set up the division: Write the dividend (the polynomial being divided) inside the division symbol and the divisor (the polynomial we are dividing by) outside.
- Divide the leading terms: Divide the leading term of the dividend by the leading term of the divisor. This result becomes the first term of the quotient.
- Multiply: Multiply the entire divisor by the first term of the quotient.
- Subtract: Subtract the result from the dividend.
- Bring down the next term: Bring down the next term from the dividend.
- Repeat: Repeat steps 2-5 until there are no more terms to bring down.
- Remainder: The final result after the last subtraction is the remainder.
Long Division Method
For a more structured approach, we often employ the long division method, which mirrors the familiar long division process for numbers. Let's break down the steps with a simple example before tackling our main problem. Imagine we want to divide (x² + 3x + 2) by (x + 1). First, we set up the division:
________
x + 1 | x² + 3x + 2
Next, we divide the leading term x² by x, which gives us x. This is the first term of our quotient. We then multiply (x + 1) by x, resulting in x² + x. Subtracting this from the dividend gives us 2x + 2. Bringing down the next term, we divide 2x by x, which gives us 2. This is the second term of our quotient. Multiplying (x + 1) by 2 gives us 2x + 2. Subtracting this leaves us with a remainder of 0. Therefore, the quotient is x + 2. Long division provides a systematic way to handle polynomial division, ensuring accuracy and clarity. By meticulously following each step, we can break down complex division problems into manageable segments, making it easier to arrive at the correct quotient and remainder.
Synthetic Division Method
Another efficient method for polynomial division is synthetic division, which is particularly useful when dividing by a linear factor of the form (x - a). Unlike long division, synthetic division focuses on the coefficients of the polynomials, making the process more streamlined. Let's illustrate this with an example. Suppose we want to divide (x³ - 4x² + 6x - 4) by (x - 2). First, we write down the coefficients of the dividend: 1, -4, 6, -4. Then, we write the root of the divisor, which is 2, to the left. We bring down the first coefficient, which is 1. We multiply 2 by 1 and write the result, 2, below the next coefficient, -4. Adding these gives us -2. We repeat the process: multiply 2 by -2 to get -4, write it below 6, and add to get 2. Finally, multiply 2 by 2 to get 4, write it below -4, and add to get 0. The last number, 0, is the remainder, and the other numbers, 1, -2, 2, are the coefficients of the quotient. Thus, the quotient is x² - 2x + 2. Synthetic division offers a faster alternative to long division, especially when the divisor is a simple linear expression. By focusing on the coefficients and streamlining the calculation process, it reduces the complexity of polynomial division, making it an indispensable tool for algebraic manipulations.
Solving the Problem: (2x³ - 2x - 12) ÷ (x - 2)
Now, let's apply these principles to the specific problem at hand: dividing (2x³ - 2x - 12) by (x - 2). This problem will allow us to put our understanding of polynomial division into practice. Dividing polynomials like this requires a systematic approach to ensure accuracy. We'll use the long division method to illustrate the process, but the underlying concepts apply to synthetic division as well.
Before we start the division, it's crucial to note that the polynomial (2x³ - 2x - 12) is missing an x² term. To maintain proper alignment during division, we need to include a placeholder term with a coefficient of zero. Thus, we rewrite the polynomial as (2x³ + 0x² - 2x - 12). This ensures that each term is aligned correctly during the long division process. Setting up the division, we have:
________
x - 2 | 2x³ + 0x² - 2x - 12
Step 1: Divide the Leading Terms
We begin by dividing the leading term of the dividend, 2x³, by the leading term of the divisor, x. This yields 2x², which becomes the first term of our quotient. This step is crucial as it sets the stage for the subsequent steps in the division process. By focusing on the leading terms, we ensure that we are reducing the degree of the dividend systematically.
2x²
x - 2 | 2x³ + 0x² - 2x - 12
Step 2: Multiply
Next, we multiply the entire divisor, (x - 2), by the first term of the quotient, 2x². This gives us 2x³ - 4x². This multiplication step is essential as it determines the portion of the dividend that will be subtracted in the next step. A careful multiplication ensures that the subtraction is performed accurately.
2x²
x - 2 | 2x³ + 0x² - 2x - 12
2x³ - 4x²
Step 3: Subtract
Now, we subtract the result, (2x³ - 4x²), from the corresponding terms of the dividend, (2x³ + 0x²). This subtraction is a critical step as it eliminates the leading term of the dividend, reducing the degree of the remaining polynomial. The result of this subtraction is 4x².
2x²
x - 2 | 2x³ + 0x² - 2x - 12
2x³ - 4x²
--------
4x²
Step 4: Bring Down the Next Term
We bring down the next term from the dividend, which is -2x. This step prepares the remaining polynomial for the next iteration of the division process. By bringing down the next term, we ensure that all parts of the dividend are considered in the division.
2x²
x - 2 | 2x³ + 0x² - 2x - 12
2x³ - 4x²
--------
4x² - 2x
Step 5: Repeat the Process
We repeat the process, dividing the leading term of the new dividend, 4x², by the leading term of the divisor, x. This yields 4x, which becomes the next term in our quotient. This repetition is the core of the long division method, allowing us to systematically reduce the dividend until we reach the remainder.
2x² + 4x
x - 2 | 2x³ + 0x² - 2x - 12
2x³ - 4x²
--------
4x² - 2x
Step 6: Multiply
We multiply the divisor, (x - 2), by 4x, giving us 4x² - 8x. This multiplication step ensures that we are subtracting the correct amount from the remaining dividend.
2x² + 4x
x - 2 | 2x³ + 0x² - 2x - 12
2x³ - 4x²
--------
4x² - 2x
4x² - 8x
Step 7: Subtract
Subtracting (4x² - 8x) from (4x² - 2x) results in 6x. This subtraction continues to reduce the degree of the remaining polynomial.
2x² + 4x
x - 2 | 2x³ + 0x² - 2x - 12
2x³ - 4x²
--------
4x² - 2x
4x² - 8x
-------
6x
Step 8: Bring Down the Next Term
We bring down the last term, -12, from the dividend. This prepares us for the final iteration of the division.
2x² + 4x
x - 2 | 2x³ + 0x² - 2x - 12
2x³ - 4x²
--------
4x² - 2x
4x² - 8x
-------
6x - 12
Step 9: Repeat
We divide 6x by x, which gives us 6. This is the final term of our quotient.
2x² + 4x + 6
x - 2 | 2x³ + 0x² - 2x - 12
2x³ - 4x²
--------
4x² - 2x
4x² - 8x
-------
6x - 12
Step 10: Multiply
Multiplying (x - 2) by 6 gives us 6x - 12. This is the final multiplication step.
2x² + 4x + 6
x - 2 | 2x³ + 0x² - 2x - 12
2x³ - 4x²
--------
4x² - 2x
4x² - 8x
-------
6x - 12
6x - 12
Step 11: Subtract
Subtracting (6x - 12) from (6x - 12) gives us a remainder of 0. This indicates that the division is exact.
2x² + 4x + 6
x - 2 | 2x³ + 0x² - 2x - 12
2x³ - 4x²
--------
4x² - 2x
4x² - 8x
-------
6x - 12
6x - 12
-------
0
Therefore, the quotient is 2x² + 4x + 6. The quotient represents the result of the polynomial division, indicating how many times the divisor goes into the dividend. In this case, 2x² + 4x + 6 is the polynomial that, when multiplied by (x - 2), yields the original polynomial (2x³ - 2x - 12). Understanding the quotient is crucial for simplifying expressions, solving equations, and analyzing the relationships between polynomials.
Conclusion
In conclusion, dividing (2x³ - 2x - 12) by (x - 2) results in the quotient 2x² + 4x + 6. This exercise underscores the importance of mastering polynomial division techniques, which are fundamental to algebraic manipulation and problem-solving. Polynomial division not only helps us simplify complex expressions but also provides valuable insights into the structure and behavior of polynomials. By understanding the steps involved in long division and synthetic division, we can confidently tackle a wide range of algebraic problems. The ability to divide polynomials effectively is a cornerstone of advanced mathematics and a crucial skill for anyone pursuing studies in science, engineering, or related fields.
Therefore, the correct answer is:
- D. 2x² + 4x + 6
