Simplifying Polynomials A Step-by-Step Guide

Introduction

In the realm of algebra, simplifying polynomial expressions is a fundamental skill. Polynomials, which are algebraic expressions consisting of variables and coefficients, can often appear complex. However, with a systematic approach, these expressions can be simplified to a more manageable form. This article delves into the process of simplifying the polynomial expression $(8j^3 + 5j^2 - 3) - (5j^3 + 7j^2 - 12j + 7)$. We will break down each step, providing a clear and comprehensive understanding of how to arrive at the simplified form. Mastering this skill is crucial for success in higher-level mathematics, as it forms the basis for solving equations, graphing functions, and more. By the end of this guide, you will be equipped with the knowledge and confidence to tackle similar polynomial simplification problems.

Understanding Polynomials

Before we dive into the simplification process, let's establish a solid understanding of what polynomials are. A polynomial is an expression containing variables (often denoted by letters like x, y, or in our case, j) and coefficients, combined using addition, subtraction, and non-negative integer exponents. Key components of a polynomial include:

• Variables: These are the unknown quantities represented by letters (e.g., j in our expression).
• Coefficients: These are the numerical values that multiply the variables (e.g., 8, 5, -3, 5, 7, -12, and 7 in our expression).
• Exponents: These indicate the power to which the variable is raised (e.g., 3 and 2 in our expression).
• Terms: These are the individual parts of the polynomial separated by addition or subtraction (e.g., $8j^3$, $5j^2$, -3, $5j^3$, $7j^2$, -12j, and 7 in our expression).

Polynomials can be classified based on the number of terms they contain:

• Monomial: A polynomial with one term (e.g., $5x^2$).
• Binomial: A polynomial with two terms (e.g., $3x + 2$).
• Trinomial: A polynomial with three terms (e.g., $x^2 - 4x + 7$).

Our expression, $(8j^3 + 5j^2 - 3) - (5j^3 + 7j^2 - 12j + 7)$, involves two trinomials. The goal of simplification is to combine like terms, which are terms that have the same variable raised to the same power. By understanding the structure of polynomials, we can approach simplification with a clear strategy.

Step 1: Distribute the Negative Sign

The first critical step in simplifying the expression $(8j^3 + 5j^2 - 3) - (5j^3 + 7j^2 - 12j + 7)$ is to address the subtraction operation between the two polynomials. Subtraction of a polynomial is equivalent to adding the negative of that polynomial. This means we need to distribute the negative sign (which can be thought of as multiplying by -1) to each term within the second set of parentheses. This process is crucial because it ensures that we correctly account for the signs of each term when combining like terms later on.

To distribute the negative sign, we perform the following operations:

• $-(5j^3) = -5j^3$
• $-(7j^2) = -7j^2$
• $-(-12j) = +12j$
• $-(7) = -7$

By distributing the negative sign, we transform the original expression into an equivalent form that is easier to work with. The expression now becomes: $8j^3 + 5j^2 - 3 - 5j^3 - 7j^2 + 12j - 7$. This step is a fundamental algebraic manipulation that allows us to remove the parentheses and prepare the expression for the next stage of simplification, which involves identifying and combining like terms. Remember, paying close attention to the signs at this stage is essential to avoid errors in the final result.

Step 2: Identify Like Terms

After distributing the negative sign, our expression stands as $8j^3 + 5j^2 - 3 - 5j^3 - 7j^2 + 12j - 7$. The next crucial step in simplifying this polynomial is to identify like terms. Like terms are those that share the same variable raised to the same power. Recognizing these terms is essential because they can be combined through addition or subtraction, which is the core of polynomial simplification. Let's break down how to identify like terms in our expression:

• Terms with $j^3$: We have $8j^3$ and $-5j^3$. These are like terms because they both contain the variable j raised to the power of 3.
• Terms with $j^2$: We have $5j^2$ and $-7j^2$. These are like terms because they both contain the variable j raised to the power of 2.
• Terms with j: We have $12j$. This is the only term with j raised to the power of 1, so it doesn't have any like terms in this expression.
• Constant Terms: We have -3 and -7. These are constant terms (numbers without any variables) and are considered like terms because they can be combined.

Identifying like terms is a foundational skill in algebra. It allows us to group together the parts of the expression that can be combined, making the simplification process more organized and efficient. By carefully examining the variables and their exponents, we can accurately pinpoint the like terms and prepare them for the next step: combining them.

Step 3: Combine Like Terms

Now that we've identified the like terms in our expression, $8j^3 + 5j^2 - 3 - 5j^3 - 7j^2 + 12j - 7$, the next step is to combine them. Combining like terms involves adding or subtracting the coefficients of those terms while keeping the variable and its exponent the same. This process effectively reduces the number of terms in the polynomial, leading us closer to the simplified form. Let's proceed with combining the like terms we identified:

• Combining $j^3$ terms: We have $8j^3$ and $-5j^3$. Adding their coefficients, we get $8 - 5 = 3$. So, $8j^3 - 5j^3 = 3j^3$.
• Combining $j^2$ terms: We have $5j^2$ and $-7j^2$. Adding their coefficients, we get $5 + (-7) = -2$. So, $5j^2 - 7j^2 = -2j^2$.
• Combining j terms: We only have one term with j, which is $12j$. Since there are no other like terms, it remains as $12j$.
• Combining Constant Terms: We have -3 and -7. Adding these, we get $-3 + (-7) = -10$.

By combining the coefficients of the like terms, we've effectively reduced the expression to a more compact form. This step is crucial in simplifying polynomials, as it consolidates the expression and makes it easier to understand and work with. The result of this step will be a simplified polynomial with fewer terms, which we will present in the next step.

Step 4: Write the Simplified Expression

After combining the like terms in the expression $8j^3 + 5j^2 - 3 - 5j^3 - 7j^2 + 12j - 7$, we have the following terms: $3j^3$, $-2j^2$, $12j$, and $-10$. The final step in simplifying the polynomial is to write the simplified expression by arranging these terms in a standard form. The standard form for a polynomial is to write the terms in descending order of their exponents. This means we start with the term with the highest exponent and proceed to the term with the lowest exponent, ending with the constant term.

Following this convention, we arrange our terms as follows:

1. Term with $j^3$: $3j^3$
2. Term with $j^2$: $-2j^2$
3. Term with j: $12j$
4. Constant term: $-10$

By arranging the terms in this order, we present the polynomial in a clear and organized manner, which is a common practice in mathematics. The simplified expression is now:

$3j^3 - 2j^2 + 12j - 10$

This is the final simplified form of the original expression $(8j^3 + 5j^2 - 3) - (5j^3 + 7j^2 - 12j + 7)$. By following the steps of distributing the negative sign, identifying like terms, combining like terms, and writing the expression in standard form, we have successfully simplified the polynomial. This simplified form is not only easier to read and understand but also more convenient for further algebraic manipulations, such as solving equations or graphing functions.

Conclusion

In this comprehensive guide, we have successfully simplified the polynomial expression $(8j^3 + 5j^2 - 3) - (5j^3 + 7j^2 - 12j + 7)$ through a step-by-step process. We began by understanding the fundamental components of polynomials, including variables, coefficients, exponents, and terms. This foundational knowledge allowed us to approach the simplification with a clear understanding of the structure we were working with.

The simplification process involved four key steps:

1. Distributing the Negative Sign: We addressed the subtraction operation by distributing the negative sign to each term within the second polynomial, ensuring that we correctly accounted for the signs of each term.
2. Identifying Like Terms: We identified like terms, which are terms that share the same variable raised to the same power. This step is crucial because it allows us to group together the parts of the expression that can be combined.
3. Combining Like Terms: We combined the like terms by adding or subtracting their coefficients, effectively reducing the number of terms in the polynomial.
4. Writing the Simplified Expression: Finally, we wrote the simplified expression in standard form, arranging the terms in descending order of their exponents.

By following these steps, we arrived at the simplified form of the polynomial: $3j^3 - 2j^2 + 12j - 10$. This simplified form is not only more concise but also easier to work with in further algebraic manipulations.

The ability to simplify polynomial expressions is a fundamental skill in algebra and is essential for success in higher-level mathematics. It forms the basis for solving equations, graphing functions, and tackling more complex algebraic problems. By mastering this skill, you equip yourself with a powerful tool that will serve you well in your mathematical journey. Remember to practice these steps with various polynomial expressions to reinforce your understanding and build confidence in your abilities. With consistent effort, you can become proficient in simplifying polynomials and excel in your mathematical endeavors.