Dividing Polynomials: A Step-by-Step Guide

Welcome! Let's dive into the world of polynomial division, a fundamental concept in algebra. In this article, we'll break down the process of dividing polynomials, step by step, making it easy to understand and apply. We'll be working through a specific example, providing a detailed walkthrough to help you master this essential skill. This skill is critical for advanced topics in mathematics such as solving for roots, graphing, and further algebraic manipulations. So, grab your pen and paper, and let's get started!

The Problem: Setting the Stage

Our task is to divide the polynomial ${x^3 - 9x - 4}$ by the polynomial ${x^2 - 3x + 2}$. This can be written as: {rac{x^3 - 9x - 4}{x^2 - 3x + 2}}. The goal is to find the quotient and the remainder. Remember, the quotient is the result of the division, and the remainder is what's left over after the division is complete. The result will look something like this: Quotient + (Remainder / Divisor).

Now, let's look at the given options: A. ${x + 3 + \frac{-2x - 10}{x^2 - 3x + 2}}$ B. ${x + \beta + \frac{-20x + 2}{x^2 - 3x + 2}}$ C. ${x - 3 + \frac{2x - 10}{x^2 - 3x + 2}}$ D. ${x - 6 + \frac{-20x + 8}{x^2 - 3x + 2}}$

Our goal is to figure out which of these options represents the correct quotient and remainder when we divide the original polynomials. We will use the method of polynomial long division to arrive at our answer, which is similar to long division with numbers, but with variables and exponents. This systematic approach ensures accuracy and clarity.

Step-by-Step Polynomial Long Division

Let's meticulously go through the polynomial long division process. Remember, we are dividing ${x^3 - 9x - 4}$ (the dividend) by ${x^2 - 3x + 2}$ (the divisor).

1. Set up the division: Write the problem like a traditional long division problem:

           ________
   

x^2 - 3x + 2 | x^3 + 0x^2 - 9x - 4 ```

Notice that we've included a ${0x^2}$ term in the dividend. This is because it is good practice to include all the terms, even if they have a zero coefficient. This ensures that the terms are correctly aligned during the division process. This is especially helpful if any terms are missing in the original polynomial.

2. Divide the leading terms: Divide the leading term of the dividend (${x^3}$) by the leading term of the divisor (${x^2}$). ${\frac{x^3}{x^2} = x}$. This becomes the first term of our quotient:

           x_______
   

x^2 - 3x + 2 | x^3 + 0x^2 - 9x - 4 ```

3. Multiply the quotient term by the divisor: Multiply the ${x}$ (from the quotient) by the entire divisor ${x^2 - 3x + 2}$: ${x * (x^2 - 3x + 2) = x^3 - 3x^2 + 2x}$. Write this result below the dividend:

           x_______
   

x^2 - 3x + 2 | x^3 + 0x^2 - 9x - 4 x^3 - 3x^2 + 2x ```

4. Subtract: Subtract the result from step 3 from the dividend. Be careful with the signs! Subtracting ${x^3 - 3x^2 + 2x}$ is the same as adding ${-x^3 + 3x^2 - 2x}$.

           x_______
   

x^2 - 3x + 2 | x^3 + 0x^2 - 9x - 4 x^3 - 3x^2 + 2x ---------- 3x^2 - 11x - 4 ```

5. Bring down the next term: Bring down the next term (there are no terms left):

6. Repeat: Now, we repeat the process. Divide the leading term of the new polynomial (${3x^2}$) by the leading term of the divisor (${x^2}$). ${\frac{3x^2}{x^2} = 3}$. This becomes the next term in the quotient:

           x + 3_____
   

x^2 - 3x + 2 | x^3 + 0x^2 - 9x - 4 x^3 - 3x^2 + 2x ---------- 3x^2 - 11x - 4 ```

7. Multiply the new quotient term by the divisor: Multiply ${3}$ (from the quotient) by the entire divisor ${x^2 - 3x + 2}$: ${3 * (x^2 - 3x + 2) = 3x^2 - 9x + 6}$. Write this result below ${3x^2 - 11x - 4}$:

           x + 3_____
   

x^2 - 3x + 2 | x^3 + 0x^2 - 9x - 4 x^3 - 3x^2 + 2x ---------- 3x^2 - 11x - 4 3x^2 - 9x + 6 ```

8. Subtract again: Subtract the result from step 7 from the previous result. Again, pay close attention to the signs!

           x + 3_____
   

x^2 - 3x + 2 | x^3 + 0x^2 - 9x - 4 x^3 - 3x^2 + 2x ---------- 3x^2 - 11x - 4 3x^2 - 9x + 6 ---------- -2x - 10 ```

9. The remainder: The resulting polynomial, ${-2x - 10}$, is the remainder. Since the degree of this polynomial (1) is less than the degree of the divisor (2), we can't continue dividing. Thus, our final answer is: ${x + 3 + \frac{-2x - 10}{x^2 - 3x + 2}}$

Analyzing the Results and Finding the Correct Answer

Now that we have performed the polynomial division, we have our quotient and remainder. We determined that the quotient is ${x + 3}$ and the remainder is ${-2x - 10}$. Thus, the correct expression representing the division is: ${x + 3 + \frac{-2x - 10}{x^2 - 3x + 2}}$. This matches option A exactly.

Let's briefly examine the other options to understand why they are incorrect. Option B includes a ${\beta}$, which is not part of the standard result of this division. Options C and D have incorrect quotients and/or incorrect remainders. The methodical breakdown of polynomial long division ensures we can arrive at the right answer every time.

By carefully working through the steps of polynomial long division, you can confidently solve these types of problems. Remember to pay close attention to signs, align the terms correctly, and double-check your calculations. Practice is key to mastering this skill! With consistent practice, you'll become proficient in dividing polynomials, making it easier to tackle more advanced algebraic problems. This skill builds a strong foundation for future topics.

Conclusion: Mastering Polynomial Division

In summary, we successfully divided ${x^3 - 9x - 4}$ by ${x^2 - 3x + 2}$ using polynomial long division. The key takeaways are the methodical approach, the attention to detail, and the understanding of the quotient and remainder. Through this process, we found the solution to be option A. Continue practicing with various polynomials to improve your understanding and proficiency. This crucial skill will be valuable in your mathematical journey.

For further learning, I suggest checking out some external resources.

Check out Khan Academy (https://www.khanacademy.org/math/algebra2/x2f8bb11595b61c86:polynomials/x2f8bb11595b61c86:div-poly/a/dividing-polynomials-with-remainder) for additional practice and in-depth explanations. They offer numerous examples and exercises that can help you solidify your understanding of polynomial division.