Polynomial Division: Solve (-3z^3 + 12z) ÷ (z - 2)

Are you struggling with polynomial division? Don't worry; you're not alone! Polynomial division can seem tricky, but with a clear explanation and step-by-step approach, you'll master it in no time. In this article, we'll walk through how to divide the polynomial (-3z^3 + 12z) by (z - 2). We’ll make sure to handle any remainders properly by expressing them as fractions. So, grab a pen and paper, and let's dive in!

Understanding Polynomial Division

Before we jump into the problem, let's quickly recap what polynomial division is all about. Polynomial division is a method used to divide one polynomial by another, and it's quite similar to the long division you learned with numbers. When we divide polynomials, we're essentially trying to find out how many times one polynomial fits into another. The key components of polynomial division include the dividend (the polynomial being divided), the divisor (the polynomial we're dividing by), the quotient (the result of the division), and the remainder (the part that's left over if the division isn't exact). Mastering polynomial division is crucial in various areas of mathematics, including algebra and calculus.

Polynomial division helps us simplify complex expressions and solve equations. Think of it as a tool that breaks down big problems into smaller, more manageable parts. You'll find it particularly useful when factoring polynomials or finding roots of polynomial equations. The process might seem intimidating at first, but with practice, it becomes second nature. Let's get started by understanding the core concepts that make polynomial division work. We will use these concepts to solve our problem step by step.

Setting Up the Division

To start, we need to set up the division problem correctly. The dividend, which is (-3z^3 + 12z), goes inside the division symbol, and the divisor, which is (z - 2), goes outside. It's crucial to make sure you've included placeholders for any missing terms in the dividend. In our case, we're missing a z^2 term, so we'll write it as 0z^2 to keep everything aligned. This step is essential for accurate calculations and prevents common mistakes. Setting up the problem correctly ensures that we can follow a systematic approach to the solution. Now, let’s see how this setup looks for our specific problem.

Performing the Long Division

The process of long division involves several steps that we repeat until we can no longer divide. First, we look at the highest degree term in the dividend (-3z^3) and the highest degree term in the divisor (z). We ask ourselves, “What do we need to multiply z by to get -3z^3?” The answer is -3z^2. We write this above the division symbol, aligned with the z^2 term. Next, we multiply the entire divisor (z - 2) by -3z^2, which gives us -3z^3 + 6z^2. We write this result below the dividend and subtract it. Remember to distribute the multiplication across all terms in the divisor. This ensures that we account for all the parts of the polynomial we're dividing by.

After subtraction, we bring down the next term from the dividend, which in this case is +12z. Now we have a new polynomial to work with: -6z^2 + 12z. We repeat the process, asking ourselves, “What do we need to multiply z by to get -6z^2?” The answer is -6z. We write this above the division symbol, aligned with the z term. We then multiply the divisor (z - 2) by -6z, which gives us -6z^2 + 12z. We write this below and subtract it. This iterative process allows us to systematically break down the polynomial and find the quotient and remainder.

Dealing with the Remainder

After subtracting -6z^2 + 12z from -6z^2 + 12z, we get 0. This means there's no remainder in this specific case. However, it's important to know how to handle remainders when they occur. If we had a remainder, we would write it as a fraction, with the remainder as the numerator and the divisor as the denominator. For example, if we had a remainder of 5, we would write it as 5/(z - 2). Including the remainder as a fraction is crucial for expressing the complete result of the division. In situations where the polynomial doesn't divide evenly, the remainder provides valuable information about the relationship between the dividend and the divisor.

Step-by-Step Solution

Now, let’s go through the step-by-step solution to divide (-3z^3 + 12z) by (z - 2):

1. Set up the long division:

                ________
   

z - 2 | -3z^3 + 0z^2 + 12z + 0 ```

Notice the inclusion of **0z^2** and **+ 0** as placeholders for the missing terms.

2. Divide the first term:

   Divide -3z^3 by z to get -3z^2.

                -3z^2
   

z - 2 | -3z^3 + 0z^2 + 12z + 0 ```

3. Multiply and subtract:

   Multiply (z - 2) by -3z^2 to get -3z^3 + 6z^2.

   Subtract this from the dividend:

                -3z^2
   

z - 2 | -3z^3 + 0z^2 + 12z + 0 -(-3z^3 + 6z^2) ------------------ -6z^2 + 12z ```

4. Bring down the next term:

   Bring down +12z.

                -3z^2
   

z - 2 | -3z^3 + 0z^2 + 12z + 0 -(-3z^3 + 6z^2) ------------------ -6z^2 + 12z ```

5. Divide the next term:

   Divide -6z^2 by z to get -6z.

                -3z^2 - 6z
   

z - 2 | -3z^3 + 0z^2 + 12z + 0 -(-3z^3 + 6z^2) ------------------ -6z^2 + 12z ```

6. Multiply and subtract:

   Multiply (z - 2) by -6z to get -6z^2 + 12z.

   Subtract this from -6z^2 + 12z:

                -3z^2 - 6z
   

z - 2 | -3z^3 + 0z^2 + 12z + 0 -(-3z^3 + 6z^2) ------------------ -6z^2 + 12z -(-6z^2 + 12z) ------------------ 0 ```

7. Final Result:

   The quotient is -3z^2 - 6z, and there is no remainder.

   Therefore, (-3z^3 + 12z) ÷ (z - 2) = -3z^2 - 6z.

Common Mistakes to Avoid

Polynomial division can be tricky, and it's easy to make mistakes if you're not careful. One common mistake is forgetting to include placeholders for missing terms in the dividend. As we saw earlier, including 0z^2 was crucial for aligning the terms correctly. Another mistake is making errors in the subtraction step. Remember to distribute the negative sign correctly when subtracting polynomials. It's also important to double-check your multiplication and division steps to ensure accuracy. Keeping track of the signs and exponents is vital for arriving at the correct solution. By being mindful of these common pitfalls, you can significantly improve your accuracy and confidence in polynomial division.

Tips for Success

To master polynomial division, practice is key. The more problems you solve, the more comfortable you'll become with the process. Start with simpler problems and gradually work your way up to more complex ones. Use online resources and textbooks to find practice problems and solutions. Additionally, breaking down each problem into smaller steps can make the process less daunting. Focus on getting each step right before moving on to the next. Remember, polynomial division is a skill that improves with consistent effort and attention to detail. Don't get discouraged by initial difficulties; persistence will pay off in the long run.

Conclusion

Congratulations! You've learned how to divide the polynomial (-3z^3 + 12z) by (z - 2). Remember, the key is to set up the problem correctly, perform the long division step by step, and handle any remainders as fractions. Polynomial division is a fundamental skill in algebra, and mastering it will help you in more advanced math courses. Keep practicing, and you'll become a pro in no time!

For further learning and practice, check out this helpful resource on Polynomial Long Division. Happy dividing!