Long Division: Find The Quotient Of (12a³-6a²-2a+7)/(4a+2)

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Hey guys! Today, we're diving into the world of polynomial long division. It might seem a bit intimidating at first, but trust me, once you get the hang of it, it's super satisfying. We're going to tackle the expression (12a³ - 6a² - 2a + 7) ÷ (4a + 2). So, grab your pencils, and let's get started!

Understanding Polynomial Long Division

Before we jump into the problem, let's quickly recap what polynomial long division is all about. Think of it like regular long division, but instead of numbers, we're dealing with polynomials. The goal is the same: to find the quotient (the result of the division) and the remainder (what's left over).

Key Concepts:

  • Dividend: The polynomial being divided (in our case, 12a³ - 6a² - 2a + 7).
  • Divisor: The polynomial we're dividing by (in our case, 4a + 2).
  • Quotient: The result of the division (what we're trying to find).
  • Remainder: The polynomial left over after the division (hopefully, it'll be zero, but sometimes it's not!).

The process involves dividing, multiplying, subtracting, and bringing down terms, just like regular long division. We'll break it down step-by-step so it's crystal clear.

Setting Up the Problem

First things first, let's set up our long division problem. Write the dividend (12a³ - 6a² - 2a + 7) inside the division bracket and the divisor (4a + 2) outside. Make sure the terms are arranged in descending order of their exponents (a³, a², a, and the constant term). This is crucial for keeping everything organized. If any terms are missing (like if there was no 'a' term), you'd want to include a placeholder with a coefficient of 0 (e.g., + 0a) to maintain the order and prevent confusion.

        ________________________
4a + 2  |  12a³ - 6a² - 2a + 7

Step-by-Step Division Process

Okay, now for the fun part – the actual division! We'll go through this step-by-step:

  1. Divide the first term:

    • Look at the first term of the dividend (12a³) and the first term of the divisor (4a).
    • Ask yourself: "What do I need to multiply 4a by to get 12a³?"
    • The answer is 3a² (because 4a * 3a² = 12a³).
    • Write 3a² above the division bracket, aligned with the a² term.
            3a²____________________

4a + 2 | 12a³ - 6a² - 2a + 7 ```

  1. Multiply:

    • Multiply the entire divisor (4a + 2) by the term we just wrote in the quotient (3a²).
    • 3a² * (4a + 2) = 12a³ + 6a²
    • Write the result (12a³ + 6a²) below the dividend, aligning like terms.
            3a²____________________

4a + 2 | 12a³ - 6a² - 2a + 7 12a³ + 6a² ```

  1. Subtract:

    • Subtract the expression we just wrote (12a³ + 6a²) from the corresponding terms in the dividend (12a³ - 6a²).
    • (12a³ - 6a²) - (12a³ + 6a²) = -12a²
    • Write the result (-12a²) below the line.
            3a²____________________

4a + 2 | 12a³ - 6a² - 2a + 7 12a³ + 6a² ----------- -12a² ```

  1. Bring down the next term:

    • Bring down the next term from the dividend (-2a) and write it next to the -12a².
    • This gives us -12a² - 2a.
            3a²____________________

4a + 2 | 12a³ - 6a² - 2a + 7 12a³ + 6a² ----------- -12a² - 2a ```

  1. Repeat the process:

    • Now, we repeat the steps with the new expression (-12a² - 2a).
    • Divide the first term (-12a²) by the first term of the divisor (4a): -12a² / 4a = -3a.
    • Write -3a in the quotient, next to 3a².
            3a² - 3a________________

4a + 2 | 12a³ - 6a² - 2a + 7 12a³ + 6a² ----------- -12a² - 2a ```

*   Multiply -3a by the divisor (4a + 2): -3a * (4a + 2) = -12a² - 6a.
*   Write the result below the -12a² - 2a, aligning like terms.

```
        3a² - 3a________________

4a + 2 | 12a³ - 6a² - 2a + 7 12a³ + 6a² ----------- -12a² - 2a -12a² - 6a ```

*   Subtract: (-12a² - 2a) - (-12a² - 6a) = 4a.
*   Write the result (4a) below the line.

```
        3a² - 3a________________

4a + 2 | 12a³ - 6a² - 2a + 7 12a³ + 6a² ----------- -12a² - 2a -12a² - 6a ----------- 4a ```

*   Bring down the next term (+7): 4a + 7.

```
        3a² - 3a________________

4a + 2 | 12a³ - 6a² - 2a + 7 12a³ + 6a² ----------- -12a² - 2a -12a² - 6a ----------- 4a + 7 ```

  1. Repeat again:

    • Divide 4a by 4a: 4a / 4a = 1.
    • Write +1 in the quotient, next to -3a.
            3a² - 3a + 1____________

4a + 2 | 12a³ - 6a² - 2a + 7 12a³ + 6a² ----------- -12a² - 2a -12a² - 6a ----------- 4a + 7 ```

*   Multiply 1 by the divisor (4a + 2): 1 * (4a + 2) = 4a + 2.
*   Write the result below the 4a + 7.

```
        3a² - 3a + 1____________

4a + 2 | 12a³ - 6a² - 2a + 7 12a³ + 6a² ----------- -12a² - 2a -12a² - 6a ----------- 4a + 7 4a + 2 ```

*   Subtract: (4a + 7) - (4a + 2) = 5.
*   Write the result (5) below the line.

```
        3a² - 3a + 1____________

4a + 2 | 12a³ - 6a² - 2a + 7 12a³ + 6a² ----------- -12a² - 2a -12a² - 6a ----------- 4a + 7 4a + 2 ------- 5 ```

  1. The End!

    • We've reached the end because the degree of the remaining term (5, which is a constant) is less than the degree of the divisor (4a + 2, which is degree 1).

The Result

So, what did we find? Well:

  • The quotient is 3a² - 3a + 1.
  • The remainder is 5.

We can write the final answer as:

3a² - 3a + 1 + 5/(4a + 2)

This means that when you divide (12a³ - 6a² - 2a + 7) by (4a + 2), you get 3a² - 3a + 1 with a remainder of 5. We express the remainder as a fraction with the divisor as the denominator.

Checking Our Work

Want to make sure we didn't make any silly mistakes? A great way to check our work is to multiply the quotient by the divisor and then add the remainder. If we did everything right, we should get back the original dividend.

(3a² - 3a + 1) * (4a + 2) + 5

Let's do it:

  • (3a² - 3a + 1) * (4a + 2) = 12a³ - 12a² + 4a + 6a² - 6a + 2 = 12a³ - 6a² - 2a + 2
  • Now, add the remainder: (12a³ - 6a² - 2a + 2) + 5 = 12a³ - 6a² - 2a + 7

Woohoo! It matches our original dividend, so we know we got it right.

Pro Tips for Polynomial Long Division

Before we wrap up, here are a few extra tips to make polynomial long division even smoother:

  • Stay Organized: Keep your terms lined up by their degrees. This makes it easier to avoid mistakes during subtraction.
  • Use Placeholders: If a term is missing in the dividend (e.g., no 'a' term), add a placeholder with a coefficient of 0 (e.g., + 0a). This helps maintain the correct order and prevents confusion.
  • Double-Check Subtraction: Subtraction is where a lot of errors happen. Pay close attention to the signs, especially when subtracting negative terms.
  • Practice Makes Perfect: Like any math skill, polynomial long division gets easier with practice. Work through plenty of examples, and you'll become a pro in no time!

Conclusion

So, there you have it! We've successfully found the quotient of (12a³ - 6a² - 2a + 7) divided by (4a + 2) using long division. Remember, the key is to take it step by step, stay organized, and double-check your work. Keep practicing, and you'll master this technique in no time. You got this!