Finding The Quotient Of (x³ + 8) ÷ (x + 2) A Step-by-Step Guide

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In the realm of mathematics, specifically within the domain of algebra, polynomial division stands as a fundamental operation. Mastering this skill unlocks the ability to simplify complex expressions, solve equations, and delve deeper into the intricacies of mathematical relationships. One common type of polynomial division involves dividing a cubic expression by a linear expression. Let's embark on a journey to understand the process of finding the quotient of (x³ + 8) ÷ (x + 2), a quintessential example that illuminates key concepts and techniques.

Decoding the Question: Setting the Stage for Polynomial Division

Before we dive into the solution, it's crucial to understand what the question is asking. The expression (x³ + 8) ÷ (x + 2) represents a division problem where the dividend is (x³ + 8) and the divisor is (x + 2). Our goal is to determine the quotient, which is the result of this division. In simpler terms, we want to find the polynomial that, when multiplied by (x + 2), yields (x³ + 8). This process is analogous to dividing numbers, where the quotient represents how many times the divisor fits into the dividend.

The expression x³ + 8 is a binomial, a polynomial with two terms. It's a special case known as the sum of cubes, which follows a specific pattern that simplifies the division process. Recognizing this pattern is a key step in efficiently solving the problem. The divisor, (x + 2), is a linear expression, a polynomial with a degree of 1. Dividing by a linear expression is a common scenario in algebra, and mastering the techniques involved is essential for success in higher-level mathematics.

To solve this problem effectively, we can employ a couple of methods: polynomial long division and the sum of cubes factorization. Each method provides a unique perspective on the problem and reinforces the underlying mathematical principles. Let's delve into each method to gain a comprehensive understanding of the solution.

Method 1: Polynomial Long Division – A Step-by-Step Approach

Polynomial long division is a systematic method for dividing polynomials, similar to the long division method used for numbers. It involves a series of steps that break down the division process into manageable parts. Let's walk through the steps involved in dividing (x³ + 8) by (x + 2):

  1. Set up the division: Write the dividend (x³ + 8) inside the division symbol and the divisor (x + 2) outside. It's crucial to include placeholder terms for any missing powers of x. In this case, we can rewrite (x³ + 8) as (x³ + 0x² + 0x + 8). This ensures that the terms are aligned correctly during the division process.

  2. Divide the leading terms: Divide the leading term of the dividend (x³) by the leading term of the divisor (x). This gives us x², which is the first term of the quotient. Write x² above the division symbol, aligning it with the x² term in the dividend.

  3. Multiply the quotient term by the divisor: Multiply x² by (x + 2), which gives us (x³ + 2x²). Write this result below the corresponding terms in the dividend.

  4. Subtract: Subtract (x³ + 2x²) from (x³ + 0x²). This gives us (-2x²). Bring down the next term from the dividend, which is 0x, resulting in (-2x² + 0x).

  5. Repeat the process: Divide the leading term of the new expression (-2x²) by the leading term of the divisor (x). This gives us (-2x), which is the next term of the quotient. Write (-2x) above the division symbol, aligning it with the x term in the dividend.

  6. Multiply and subtract: Multiply (-2x) by (x + 2), which gives us (-2x² - 4x). Write this result below (-2x² + 0x) and subtract. This gives us (4x). Bring down the last term from the dividend, which is 8, resulting in (4x + 8).

  7. Final step: Divide the leading term of the new expression (4x) by the leading term of the divisor (x). This gives us 4, which is the final term of the quotient. Write 4 above the division symbol, aligning it with the constant term in the dividend.

  8. Multiply and subtract: Multiply 4 by (x + 2), which gives us (4x + 8). Write this result below (4x + 8) and subtract. This gives us a remainder of 0.

Therefore, the quotient of (x³ + 8) ÷ (x + 2) obtained through polynomial long division is x² - 2x + 4. This method provides a clear and structured way to perform polynomial division, especially for more complex expressions.

Method 2: Leveraging the Sum of Cubes Factorization

An alternative and often more efficient method for solving this problem involves recognizing and utilizing the sum of cubes factorization pattern. The sum of cubes formula states that: a³ + b³ = (a + b)(a² - ab + b²). This formula provides a direct way to factor expressions in the form of a sum of cubes, which can greatly simplify division problems.

In our case, (x³ + 8) can be viewed as a sum of cubes, where a = x and b = 2 (since 8 = 2³). Applying the sum of cubes formula, we can factor (x³ + 8) as follows:

x³ + 8 = x³ + 2³ = (x + 2)(x² - 2x + 2²)

Simplifying further, we get:

x³ + 8 = (x + 2)(x² - 2x + 4)

Now, we can rewrite the original division problem as:

(x³ + 8) ÷ (x + 2) = [(x + 2)(x² - 2x + 4)] ÷ (x + 2)

Since (x + 2) appears in both the numerator and the denominator, we can cancel them out, leaving us with:

(x² - 2x + 4)

Thus, using the sum of cubes factorization, we arrive at the same quotient: x² - 2x + 4. This method showcases the power of pattern recognition in mathematics and provides a more concise approach compared to long division in this specific scenario.

The Quotient Unveiled: x² - 2x + 4

Both polynomial long division and the sum of cubes factorization methods have led us to the same answer: the quotient of (x³ + 8) ÷ (x + 2) is x² - 2x + 4. This result is a quadratic expression, a polynomial with a degree of 2. The process of dividing a cubic expression by a linear expression often results in a quadratic quotient, as seen in this example.

The quotient x² - 2x + 4 represents the polynomial that, when multiplied by the divisor (x + 2), yields the dividend (x³ + 8). We can verify this by performing the multiplication:

(x + 2)(x² - 2x + 4) = x(x² - 2x + 4) + 2(x² - 2x + 4)

Expanding the expression, we get:

x³ - 2x² + 4x + 2x² - 4x + 8

Simplifying by combining like terms, we obtain:

x³ + 8

This confirms that our quotient, x² - 2x + 4, is indeed the correct result. The ability to verify solutions is a crucial aspect of mathematical problem-solving, ensuring accuracy and building confidence in one's understanding.

Selecting the Correct Answer: Option B

Now that we have determined the quotient to be x² - 2x + 4, we can confidently select the correct answer from the given options. By comparing our result with the options provided, we can clearly see that:

  • A. x² + 2x + 4 - Incorrect
  • B. x² - 2x + 4 - Correct
  • C. x² + 4 - Incorrect
  • D. x² - 4 - Incorrect

Therefore, the correct answer is Option B. This final step reinforces the importance of careful comparison and attention to detail in selecting the appropriate solution.

Mastering Polynomial Division: Key Takeaways and Further Exploration

Dividing polynomials, as demonstrated in finding the quotient of (x³ + 8) ÷ (x + 2), is a cornerstone skill in algebra. Through this exploration, we have reinforced several key concepts and techniques:

  • Polynomial Long Division: A systematic method for dividing polynomials, providing a step-by-step approach to break down complex problems.
  • Sum of Cubes Factorization: Recognizing and applying the sum of cubes formula (a³ + b³ = (a + b)(a² - ab + b²)) can significantly simplify division problems involving expressions in this form.
  • Verification: Always verify your solution by multiplying the quotient by the divisor to ensure it equals the dividend. This step is crucial for accuracy and building confidence.
  • Pattern Recognition: Identifying patterns, such as the sum of cubes, can lead to more efficient problem-solving strategies.

To further enhance your understanding of polynomial division, consider exploring the following:

  • Practice more examples of polynomial long division with varying degrees and coefficients.
  • Explore other factorization patterns, such as the difference of squares and the difference of cubes.
  • Investigate the relationship between polynomial division and the Remainder Theorem and the Factor Theorem.
  • Consider how polynomial division is applied in more advanced mathematical concepts, such as finding roots of polynomials and solving rational equations.

By continuously practicing and exploring these concepts, you will solidify your understanding of polynomial division and unlock new avenues for mathematical exploration. The journey of learning mathematics is a continuous process of discovery, and mastering fundamental skills like polynomial division is a crucial step in that journey.