Solve The Equation (-6s^2 + 12s - 8) - (3s^2 + 8s - 6).

Polynomial subtraction can seem daunting at first glance, but with a systematic approach and a clear understanding of the underlying principles, it can become a manageable and even enjoyable mathematical exercise. In this comprehensive guide, we will delve into the intricacies of polynomial subtraction, equipping you with the knowledge and skills to confidently tackle any problem that comes your way. We will dissect the given problem, $\left(-6s^2 + 12s - 8\right) - \left(3s^2 + 8s - 6\right) =$, exploring the concepts and techniques involved in arriving at the correct solution. Whether you are a student grappling with algebraic expressions or simply seeking to refresh your mathematical prowess, this article will serve as your trusted companion in the world of polynomial subtraction.

Deconstructing the Polynomial Expression

The cornerstone of polynomial subtraction lies in the ability to manipulate algebraic expressions with precision and accuracy. Before we embark on the subtraction process, it is crucial to understand the anatomy of a polynomial. A polynomial is essentially an expression comprising variables and coefficients, connected by mathematical operations such as addition, subtraction, and multiplication. The variables in a polynomial are raised to non-negative integer powers, and the coefficients are constants that multiply the variable terms. In our given problem, we encounter two polynomials: $\left(-6s^2 + 12s - 8\right)$ and $\left(3s^2 + 8s - 6\right)$. The variable in both polynomials is 's', and the coefficients are the numerical values associated with each term. For instance, in the first polynomial, the coefficient of the $s^2$ term is -6, the coefficient of the 's' term is 12, and the constant term is -8. Similarly, in the second polynomial, the coefficients are 3, 8, and -6 for the $s^2$, 's', and constant terms, respectively. A firm grasp of these fundamental concepts is paramount to successfully navigating the realm of polynomial subtraction.

Understanding the Subtraction Process

The heart of polynomial subtraction lies in the distribution of the negative sign. When subtracting one polynomial from another, we are essentially multiplying the second polynomial by -1 and then adding the result to the first polynomial. This critical step ensures that we correctly account for the signs of each term in the second polynomial. In our problem, $\left(-6s^2 + 12s - 8\right) - \left(3s^2 + 8s - 6\right)$, we begin by distributing the negative sign to each term within the second set of parentheses. This transforms the expression into $\left(-6s^2 + 12s - 8\right) + \left(-3s^2 - 8s + 6\right)$. Notice how the signs of each term in the second polynomial have been flipped. The positive $3s^2$ becomes negative $-3s^2$, the positive 8s becomes negative -8s, and the negative -6 becomes positive +6. This distribution of the negative sign is a pivotal step in the subtraction process, paving the way for the next stage of combining like terms.

Combining Like Terms: The Key to Simplification

Once we have successfully distributed the negative sign, the next crucial step is to combine like terms. Like terms are those that share the same variable and exponent. In other words, they have the same variable raised to the same power. In our transformed expression, $\left(-6s^2 + 12s - 8\right) + \left(-3s^2 - 8s + 6\right)$, we can identify three pairs of like terms: $-6s^2$ and $-3s^2$ (both $s^2$ terms), 12s and -8s (both 's' terms), and -8 and 6 (both constant terms). To combine like terms, we simply add their coefficients. For the $s^2$ terms, we have -6 + (-3) = -9, resulting in $-9s^2$. For the 's' terms, we have 12 + (-8) = 4, resulting in 4s. And for the constant terms, we have -8 + 6 = -2. By combining these like terms, we simplify the expression, bringing us closer to the final answer. The ability to identify and combine like terms is a fundamental skill in algebra, enabling us to reduce complex expressions to their simplest forms.

Step-by-Step Solution

Now that we have laid the groundwork by understanding the concepts and techniques involved, let's walk through the step-by-step solution to the problem $\left(-6s^2 + 12s - 8\right) - \left(3s^2 + 8s - 6\right) =$. This will solidify your understanding and demonstrate how to apply the learned principles in a practical setting.

1. Distribute the Negative Sign: As we discussed earlier, the first step is to distribute the negative sign to each term within the second set of parentheses: $\left(-6s^2 + 12s - 8\right) - \left(3s^2 + 8s - 6\right) = -6s^2 + 12s - 8 - 3s^2 - 8s + 6$

2. Identify Like Terms: Next, we identify the terms that share the same variable and exponent:

   • $s^2$ terms: $-6s^2$ and $-3s^2$
   • 's' terms: 12s and -8s
   • Constant terms: -8 and 6

3. Combine Like Terms: Now, we add the coefficients of the like terms:

   • $s^2$ terms: $-6s^2 + (-3s^2) = -9s^2$
   • 's' terms: $12s + (-8s) = 4s$
   • Constant terms: $-8 + 6 = -2$

4. Write the Simplified Expression: Finally, we combine the results to form the simplified expression: $-9s^2 + 4s - 2$

Therefore, the solution to the problem $\left(-6s^2 + 12s - 8\right) - \left(3s^2 + 8s - 6\right) =$ is $-9s^2 + 4s - 2$, which corresponds to option B.

Common Pitfalls to Avoid

While polynomial subtraction is a straightforward process when executed correctly, there are several common pitfalls that students often encounter. Being aware of these potential errors can help you avoid them and ensure accuracy in your calculations. One of the most frequent mistakes is failing to distribute the negative sign properly. Remember that the negative sign applies to every term within the parentheses being subtracted, not just the first term. Another common error is misidentifying like terms. It is crucial to carefully examine the variables and exponents to ensure that you are combining only terms that share the same variable raised to the same power. For instance, $s^2$ and 's' are not like terms and cannot be combined. Additionally, sign errors are a common culprit in polynomial subtraction. Pay close attention to the signs of the coefficients when adding or subtracting like terms. A simple sign mistake can lead to an incorrect final answer. By being mindful of these common pitfalls and taking extra care in your calculations, you can significantly reduce the likelihood of errors in polynomial subtraction.

Conclusion: Mastering Polynomial Subtraction

In conclusion, polynomial subtraction is a fundamental concept in algebra that can be mastered with a clear understanding of the underlying principles and consistent practice. By meticulously distributing the negative sign, accurately identifying and combining like terms, and avoiding common pitfalls, you can confidently navigate the world of polynomial subtraction. The step-by-step solution presented in this guide provides a solid framework for tackling polynomial subtraction problems, and the insights into common errors will help you refine your approach and enhance your accuracy. So, embrace the challenge of polynomial subtraction, hone your skills, and unlock the power of algebraic expressions.