Solve Polynomials: A Step-by-Step Subtraction Guide

Hey there, math enthusiasts! Today, we're diving into the fascinating world of polynomial subtraction. Polynomials might seem intimidating at first glance, but trust me, they're like puzzles waiting to be solved. We'll break down the expression (0. 4k³ - 2.5k) - (2.4k³ + 3k² - 1.2k) step-by-step, making sure you grasp every detail. So, grab your pencils, and let's get started!

Understanding the Basics of Polynomials

Before we jump into the problem, let's quickly recap what polynomials are. A polynomial is essentially an expression containing variables (like 'k' in our case) raised to different powers, combined with constants and coefficients. Think of them as building blocks of algebraic expressions. Each part of the polynomial that includes a coefficient, variable, and exponent is called a term. For example, in the polynomial 0.4k³ - 2.5k, we have two terms: 0.4k³ and -2.5k. The first term has a coefficient of 0.4, a variable 'k', and an exponent of 3. The second term has a coefficient of -2.5, a variable 'k', and an exponent of 1 (which is usually not explicitly written). When subtracting polynomials, the key is to combine like terms. Like terms are those that have the same variable raised to the same power. For example, 3x² and -5x² are like terms because they both have the variable 'x' raised to the power of 2. However, 3x² and 3x are not like terms because the exponents are different. Similarly, 2y³ and 7y³ are like terms, while 2y³ and 2y² are not. This concept of like terms is crucial because we can only add or subtract terms that are like each other. Trying to combine unlike terms is like trying to add apples and oranges – it just doesn't work! So, keep this in mind as we move forward and tackle the problem at hand. Remember, the goal is to simplify the expression by identifying and combining those terms that share the same variable and exponent.

Step 1: Distributing the Negative Sign

The first key step in simplifying the expression (0.4k³ - 2.5k) - (2.4k³ + 3k² - 1.2k) is to distribute the negative sign in front of the second parenthesis. This is a crucial step because it changes the signs of each term inside the parenthesis, which directly impacts the final result. When you distribute a negative sign, you're essentially multiplying each term inside the parenthesis by -1. So, the expression (2.4k³ + 3k² - 1.2k) becomes -2.4k³ - 3k² + 1.2k. Think of it like this: the negative sign is like a little agent of change, flipping the signs of everything it touches. A positive term becomes negative, and a negative term becomes positive. This process is crucial because it ensures that we correctly account for the subtraction operation. Many mistakes in polynomial subtraction occur because this step is skipped or performed incorrectly. So, take your time and be extra careful when distributing the negative sign. Double-check each term to make sure you've flipped the sign correctly. Once you've distributed the negative sign, the original expression transforms into 0.4k³ - 2.5k - 2.4k³ - 3k² + 1.2k. This new form of the expression is much easier to work with because we've eliminated the parenthesis and can now proceed with combining like terms.

Step 2: Identifying and Grouping Like Terms

Now that we've distributed the negative sign, the expression looks like this: 0.4k³ - 2.5k - 2.4k³ - 3k² + 1.2k. The next step is to identify and group like terms. Remember, like terms are those that have the same variable raised to the same power. In our expression, we have three different types of terms: k³ terms, k² terms, and k terms. Let's start by identifying the k³ terms. We have 0.4k³ and -2.4k³. These are like terms because they both have the variable 'k' raised to the power of 3. Next, let's look for the k² terms. We only have one k² term in this expression: -3k². Since there are no other terms with k², it will remain as is for now. Finally, let's identify the k terms. We have -2.5k and +1.2k. These are like terms because they both have the variable 'k' raised to the power of 1 (which is usually not explicitly written). Now that we've identified the like terms, let's group them together. This will make it easier to combine them in the next step. We can rearrange the expression to group the like terms next to each other: 0. 4k³ - 2.4k³ - 3k² - 2.5k + 1.2k. Notice how we've simply rearranged the terms without changing their signs. This grouping step is a matter of organization, making the subsequent calculations smoother and less prone to errors. By grouping like terms, we visually set the stage for the final act of combining them. It's like sorting puzzle pieces before assembling them – it might seem like a small step, but it makes the overall process much more efficient.

Step 3: Combining Like Terms

We've successfully grouped our like terms, and now it's time for the most satisfying part: combining them! We have our expression neatly arranged as 0.4k³ - 2.4k³ - 3k² - 2.5k + 1.2k. Remember, we can only add or subtract terms that are alike, meaning they have the same variable raised to the same power. Let's start with the k³ terms: 0.4k³ - 2.4k³. To combine these, we simply add or subtract their coefficients. In this case, we have 0.4 - 2.4, which equals -2. So, 0.4k³ - 2.4k³ simplifies to -2k³. Next, let's move on to the k² terms. We only have one k² term, which is -3k². Since there's nothing to combine it with, it remains as -3k². Now, let's tackle the k terms: -2.5k + 1.2k. Again, we combine their coefficients: -2.5 + 1.2, which equals -1.3. So, -2.5k + 1.2k simplifies to -1.3k. Finally, we put all the simplified terms together to get our final answer. We have -2k³ from the k³ terms, -3k² from the k² term, and -1.3k from the k terms. So, our simplified expression is -2k³ - 3k² - 1.3k. And there you have it! We've successfully navigated the polynomial subtraction and arrived at the simplified answer. This process might seem lengthy when explained in detail, but with practice, you'll be zipping through these problems like a pro. Remember, the key is to distribute the negative sign carefully, identify and group like terms, and then combine them accurately. With these steps in mind, you'll conquer any polynomial subtraction that comes your way!

Final Answer and Conclusion

So, after carefully working through each step, we've arrived at our final, simplified expression: -2k³ - 3k² - 1.3k. This is the result of subtracting (2.4k³ + 3k² - 1.2k) from (0.4k³ - 2.5k). Let's recap the key steps we took to get here:

1. We started by distributing the negative sign, which is crucial for correctly subtracting polynomials.
2. Then, we identified and grouped like terms, making it easier to see which terms could be combined.
3. Finally, we combined the like terms by adding or subtracting their coefficients.

By following these steps, we transformed a seemingly complex expression into a much simpler form. Polynomial subtraction might have seemed daunting at first, but hopefully, this step-by-step guide has demystified the process. Remember, math is like learning a new language – it takes practice, patience, and a willingness to break things down into smaller, manageable steps. Keep practicing, and you'll find that these types of problems become second nature. And most importantly, don't be afraid to make mistakes! Mistakes are learning opportunities in disguise. So, embrace the challenge, keep exploring, and happy problem-solving!

Therefore, the correct answer is -2k³ - 3k² - 1.3k