Simplifying Polynomial Expressions A Step-by-Step Guide

Polynomial expressions are fundamental in algebra, and the ability to simplify them is a crucial skill for anyone studying mathematics. This article delves into the process of simplifying a specific polynomial expression, providing a step-by-step explanation to enhance understanding and proficiency. Let's explore the intricacies of simplifying the expression $(3 - 13x - 7x^2) - (5x^2 + 12x - 10)$.

Understanding the Basics of Polynomials

Before we dive into the simplification process, it's essential to grasp the fundamental concepts of polynomials. Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Each term in a polynomial is a monomial, which is a product of a constant and variables raised to non-negative integer powers. For instance, in the given expression, $-7x^2$, $-13x$, and $3$ are all monomials.

The degree of a polynomial is the highest power of the variable in the expression. In the polynomial $(3 - 13x - 7x^2)$, the highest power of $x$ is 2, so the degree of this polynomial is 2. Similarly, the degree of the polynomial $(5x^2 + 12x - 10)$ is also 2. Understanding the degree of a polynomial is crucial for organizing and simplifying expressions effectively.

When simplifying polynomials, the primary goal is to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, $-7x^2$ and $5x^2$ are like terms because they both have the variable $x$ raised to the power of 2. Similarly, $-13x$ and $12x$ are like terms because they both have the variable $x$ raised to the power of 1. Constant terms, such as $3$ and $-10$, are also like terms.

To simplify a polynomial expression, we first need to identify the like terms and then combine them by adding or subtracting their coefficients. This process involves careful attention to signs and the order of operations. By mastering these basic concepts, you'll be well-prepared to tackle more complex polynomial expressions.

Step-by-Step Simplification Process

Now, let's break down the simplification of the expression $(3 - 13x - 7x^2) - (5x^2 + 12x - 10)$ step by step. This process involves distributing the negative sign, identifying like terms, and combining them.

1. Distribute the Negative Sign

The first step in simplifying the expression is to distribute the negative sign in front of the second parenthesis. This means we need to multiply each term inside the second parenthesis by $-1$. The expression then becomes:

$3 - 13x - 7x^2 - 5x^2 - 12x + 10$

Distributing the negative sign correctly is crucial because it changes the signs of the terms inside the parenthesis. A common mistake is to only change the sign of the first term, which leads to an incorrect simplification. By carefully distributing the negative sign, we ensure that all terms are accounted for and the expression is ready for the next step.

2. Identify Like Terms

The next step is to identify the like terms in the expression. As we discussed earlier, like terms are terms that have the same variable raised to the same power. In our expression, the like terms are:

• $-7x^2$ and $-5x^2$ (terms with $x^2$)
• $-13x$ and $-12x$ (terms with $x$)
• $3$ and $10$ (constant terms)

Identifying like terms is a critical step because it allows us to group the terms that can be combined. This makes the simplification process more organized and reduces the likelihood of errors. By clearly identifying the like terms, we set the stage for the final step of combining them.

3. Combine Like Terms

The final step is to combine the like terms by adding or subtracting their coefficients. Let's combine the like terms we identified in the previous step:

• Combine the $x^2$ terms: $-7x^2 - 5x^2 = -12x^2$
• Combine the $x$ terms: $-13x - 12x = -25x$
• Combine the constant terms: $3 + 10 = 13$

Now, we can write the simplified expression by combining these results:

$-12x^2 - 25x + 13$

This is the simplified form of the original expression. By carefully combining like terms, we have reduced the expression to its simplest form, making it easier to work with in further algebraic manipulations.

Evaluating the Options

Now that we have simplified the expression, let's evaluate the given options to determine the correct answer.

The original question asks for the simplified form of the expression $(3 - 13x - 7x^2) - (5x^2 + 12x - 10)$. We have determined that the simplified form is $-12x^2 - 25x + 13$. Let's compare this result with the options provided:

• A. $3x^2 - 25x - 2$
• B. $-12x^2 - 25x + 13$
• C. $-12x^2 - x - 7$
• D. $-12x^2 - 25x - 7$

By comparing our simplified expression with the options, we can see that option B, $-12x^2 - 25x + 13$, matches our result. Therefore, option B is the correct answer.

Common Mistakes to Avoid

Simplifying polynomial expressions can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

1. Forgetting to Distribute the Negative Sign

As we discussed earlier, forgetting to distribute the negative sign is a common mistake. When subtracting one polynomial from another, it's crucial to multiply each term in the second polynomial by $-1$. Failing to do so will result in an incorrect simplification.

2. Combining Unlike Terms

Another common mistake is combining terms that are not like terms. Remember, like terms have the same variable raised to the same power. You cannot combine terms like $-7x^2$ and $-13x$ because they have different powers of $x$. Only combine terms that have the same variable and exponent.

3. Sign Errors

Sign errors are also prevalent when simplifying expressions. Pay close attention to the signs of the terms, especially when distributing the negative sign and combining like terms. A simple sign error can lead to an incorrect answer. Double-check your work to ensure that you have the correct signs for each term.

4. Order of Operations

Following the correct order of operations is essential when simplifying expressions. Make sure to distribute any negative signs before combining like terms. Ignoring the order of operations can lead to errors in your simplification.

5. Careless Arithmetic

Finally, careless arithmetic mistakes can also lead to incorrect answers. Take your time when adding and subtracting coefficients, and double-check your work to avoid simple arithmetic errors.

By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in simplifying polynomial expressions.

Practice Problems

To further enhance your understanding and skills in simplifying polynomial expressions, it's essential to practice with various problems. Here are a few practice problems for you to try:

1. Simplify: $(4x^2 - 7x + 2) - (2x^2 + 3x - 5)$
2. Simplify: $(5x^3 + 2x^2 - x + 8) + (3x^3 - 4x^2 + 6x - 1)$
3. Simplify: $(9 - 2x + 6x^2) - (4x^2 - 5x + 1)$
4. Simplify: $(7x^2 - 3x + 1) + (2x^2 + 8x - 4) - (x^2 - x + 3)

Working through these practice problems will help you solidify your understanding of the simplification process and improve your problem-solving skills. Remember to follow the steps we discussed earlier: distribute any negative signs, identify like terms, and combine them. By practicing regularly, you'll become more proficient in simplifying polynomial expressions.

Conclusion

In this article, we have explored the process of simplifying polynomial expressions, focusing on the specific example of $(3 - 13x - 7x^2) - (5x^2 + 12x - 10)$. We broke down the simplification process into manageable steps, including distributing the negative sign, identifying like terms, and combining them. By following these steps carefully, we arrived at the simplified form of the expression: $-12x^2 - 25x + 13$.

We also discussed common mistakes to avoid, such as forgetting to distribute the negative sign, combining unlike terms, and making sign errors. By being aware of these potential pitfalls, you can improve your accuracy and avoid making these mistakes in your own work.

Finally, we provided practice problems to help you further develop your skills in simplifying polynomial expressions. Remember, practice is key to mastering any mathematical concept. By working through various problems, you'll become more confident and proficient in simplifying expressions.

Simplifying polynomial expressions is a fundamental skill in algebra, and it's essential for success in more advanced mathematical topics. By understanding the concepts and following the steps outlined in this article, you can confidently simplify polynomial expressions and excel in your math studies.