Simplify (-x²-4x+1)-(2x²+9): A Step-by-Step Guide

Hey guys! Ever found yourself staring at a polynomial subtraction problem and feeling a bit lost? Don't worry; you're not alone! Polynomial subtraction might seem daunting at first, but with a clear understanding of the underlying concepts and a step-by-step approach, it becomes a breeze. In this article, we'll tackle the expression $\left(-x^2-4 x+1\right)-\left(2 x^2+9\right)$, breaking it down into manageable steps. We'll not only solve this specific problem but also equip you with the knowledge to confidently handle any polynomial subtraction that comes your way. So, let's dive in and unravel the mysteries of polynomial subtraction together!

Understanding Polynomials: The Building Blocks

Before we jump into the subtraction itself, let's quickly recap what polynomials are. A polynomial is essentially an expression made up of variables (like x) and coefficients (the numbers in front of the variables), combined using addition, subtraction, and non-negative integer exponents (powers). Think of it as a mathematical sentence composed of terms. For instance, in our expression $\left(-x^2-4 x+1\right)-\left(2 x^2+9\right)$, we have two polynomials: $-x^2 - 4x + 1$ and $2x^2 + 9$. Each of these is a polynomial expression.

The individual parts of a polynomial that are added or subtracted are called terms. In the polynomial $-x^2 - 4x + 1$, the terms are $-x^2$, $-4x$, and $1$. It's important to pay attention to the signs (positive or negative) in front of each term, as these signs are crucial when performing operations like subtraction. The coefficients are the numerical factors multiplying the variables. For example, in the term $-x^2$, the coefficient is -1 (since $-x^2$ is the same as $-1 * x^2$), and in the term $-4x$, the coefficient is -4. The degree of a term is the exponent of the variable. In $-x^2$, the degree is 2, and in $-4x$, the degree is 1 (since $x$ is the same as $x^1$). Constant terms, like the 1 and 9 in our expression, have a degree of 0 (think of them as being multiplied by $x^0$, which is equal to 1).

Understanding these basic components – terms, coefficients, and degrees – is fundamental to mastering polynomial operations. It's like knowing the alphabet before you can write words. Once you're comfortable with these building blocks, you'll find polynomial subtraction (and other polynomial operations) much less intimidating. We will also learn about like terms in the next section, which is a crucial concept for polynomial subtraction.

The Key to Subtraction: Identifying and Combining Like Terms

The secret to smoothly subtracting polynomials lies in the concept of like terms. Like terms are terms that have the same variable raised to the same power. Think of them as belonging to the same "family" of terms. For example, $-x^2$ and $2x^2$ are like terms because they both have the variable x raised to the power of 2. Similarly, $-4x$ and, say, $3x$ would be like terms because they both have x raised to the power of 1 (or just x). However, $-x^2$ and $-4x$ are not like terms because they have different powers of x. One has $x^2$, and the other has x. Constant terms, like 1 and 9, are also considered like terms because they both have a degree of 0 (they can be thought of as having $x^0$ which equals 1).

Why are like terms so important for subtraction? Because we can only directly add or subtract like terms. It's like adding apples to apples or oranges to oranges – you can't directly add an apple to an orange and get a meaningful single answer. Similarly, we can combine the coefficients of like terms, but we can't combine terms with different variables or exponents. For instance, we can combine $-x^2$ and $2x^2$, but we can't directly combine $-x^2$ and $-4x$. When subtracting polynomials, our goal is to identify like terms within the expressions and then combine them appropriately.

In our example, $\left(-x^2-4 x+1\right)-\left(2 x^2+9\right)$, we need to identify the like terms in both polynomials. In the first polynomial, $-x^2 - 4x + 1$, we have the terms $-x^2$, $-4x$, and 1. In the second polynomial, $2x^2 + 9$, we have the terms $2x^2$ and 9. Now we can see the like terms more clearly: $-x^2$ and $2x^2$ are like terms (both have $x^2$), and 1 and 9 are like terms (both are constants). The term $-4x$ in the first polynomial doesn't have a like term in the second polynomial, so it will remain as is in our final result. Recognizing these like terms is the critical step before we actually perform the subtraction. Once we've identified them, we're ready to move on to the next step: distributing the negative sign.

Distributing the Negative: The Key to Correct Subtraction

Now that we understand like terms, let's tackle a crucial step in polynomial subtraction: distributing the negative sign. Remember, when we subtract one polynomial from another, we're essentially subtracting each term of the second polynomial. This is where the distributive property comes into play. The distributive property states that a(b + c) = ab + ac. In our case, the "a" is the negative sign (which is like -1), and the "(b + c)" is the second polynomial.

So, to subtract $\left(2 x^2+9\right)$ from $\left(-x^2-4 x+1\right)$, we need to distribute the negative sign to each term inside the parentheses of the second polynomial. This means we multiply each term in $\left(2 x^2+9\right)$ by -1. When we multiply $2x^2$ by -1, we get $-2x^2$. And when we multiply 9 by -1, we get -9. So, the expression $\left(2 x^2+9\right)$ becomes $-2x^2 - 9$ after distributing the negative sign. Think of it like changing the signs of each term inside the parentheses.

This step is absolutely crucial for getting the correct answer in polynomial subtraction. Forgetting to distribute the negative sign is a very common mistake, and it will lead to an incorrect result. By carefully distributing the negative sign, we transform the subtraction problem into an addition problem, which is often easier to handle. Our original expression, $\left(-x^2-4 x+1\right)-\left(2 x^2+9\right)$, now becomes $\left(-x^2-4 x+1\right) + \left(-2 x^2-9\right)$. See how we've changed the subtraction to addition by distributing the negative? Now, we're ready for the final step: combining the like terms we identified earlier.

Distributing the negative sign ensures we're subtracting the entire second polynomial, not just the first term. It's a small step, but it has a big impact on the accuracy of our solution. So, always remember to distribute that negative sign before you start combining like terms! With this step mastered, we're one step closer to solving our polynomial subtraction problem.

Combining Like Terms: The Final Showdown

Alright, guys, we've reached the final stage! We've identified like terms, distributed the negative sign, and now it's time to combine those like terms and simplify our expression. Remember, combining like terms means adding or subtracting the coefficients of the terms that have the same variable raised to the same power.

After distributing the negative sign, our expression looks like this: $\left(-x^2-4 x+1\right) + \left(-2 x^2-9\right)$. Now, let's group the like terms together to make the process clearer. We have the $x^2$ terms: $-x^2$ and $-2x^2$. We have the constant terms: 1 and -9. And we have the term $-4x$, which doesn't have a like term in the second polynomial.

Let's start with the $x^2$ terms. We have $-x^2 + (-2x^2)$. Remember that $-x^2$ is the same as $-1x^2$. So, we're adding the coefficients -1 and -2, which gives us -3. Therefore, $-x^2 + (-2x^2) = -3x^2$. Moving on to the constant terms, we have 1 + (-9). Adding 1 and -9 gives us -8. So, 1 + (-9) = -8. Finally, we have the term $-4x$, which doesn't have a like term to combine with. It simply stays as $-4x$ in our simplified expression.

Now, let's put everything together. We have $-3x^2$ from combining the $x^2$ terms, $-4x$ which remained as is, and -8 from combining the constant terms. So, our final simplified expression is $-3x^2 - 4x - 8$. That's it! We've successfully subtracted the polynomials. By carefully identifying like terms, distributing the negative sign, and combining the like terms, we arrived at the solution.

Solution and Summary: Putting It All Together

So, the result of performing the operation $\left(-x^2-4 x+1\right)-\left(2 x^2+9\right)$ is -3x² - 4x - 8. Let's quickly recap the steps we took to get there. First, we understood the basics of polynomials and like terms. Then, we identified the like terms in the expression. Next, we distributed the negative sign to each term in the second polynomial, which is a crucial step to avoid errors. Finally, we combined the like terms by adding their coefficients to arrive at our simplified answer.

This process might seem like a lot of steps, but with practice, it becomes second nature. The key is to be methodical and pay attention to detail. Always remember to distribute the negative sign and to combine only like terms. Polynomial subtraction is a fundamental skill in algebra, and mastering it will open doors to more advanced concepts.

Congratulations, guys! You've successfully navigated the world of polynomial subtraction. Remember to practice regularly, and don't be afraid to tackle more complex problems. The more you practice, the more confident you'll become. Keep up the great work, and happy calculating!

Practice Problems: Sharpen Your Skills

To truly master polynomial subtraction, practice is key! Here are a few practice problems for you to try. Work through them using the steps we've discussed, and check your answers. Don't worry if you make mistakes – that's how we learn! The important thing is to keep practicing and building your skills.

1. (3x² + 2x - 1) - (x² - 5x + 4)
2. (4y³ - 2y + 7) - (2y³ + 3y² - y)
3. (-2a² + 5a - 3) - (-a² + 4)

Try these problems on your own, and then check your answers. If you get stuck, review the steps we discussed earlier in the article. Remember to identify like terms, distribute the negative sign, and combine like terms carefully.

Further Learning: Expanding Your Knowledge

If you're eager to delve deeper into the world of polynomials, there are many resources available to you. Textbooks, online tutorials, and educational websites can provide more examples, explanations, and practice problems. Consider exploring topics like polynomial multiplication, division, and factoring to further expand your knowledge. The more you learn about polynomials, the more confident you'll become in your algebraic abilities.

Polynomials are a fundamental concept in mathematics, and understanding them is essential for success in algebra and beyond. So, keep learning, keep practicing, and keep exploring the fascinating world of math!