Solving Polynomial Expressions: A Step-by-Step Guide

Polynomial expressions can seem daunting at first glance, especially when they involve multiple terms and exponents. But don't worry! With a systematic approach, you can easily perform the indicated operations and simplify these expressions. In this guide, we'll walk through an example problem step by step, explaining the key concepts and techniques along the way. Understanding polynomial expressions is crucial in various fields like algebra, calculus, and even real-world applications such as physics and engineering. This guide aims to provide a clear and comprehensive understanding of how to manipulate and simplify these expressions. We'll break down each step, making it easy to follow along and apply to similar problems. Whether you're a student tackling homework or someone looking to brush up on your math skills, this guide is here to help.

Problem: Simplifying $3 x^2(5 x^3+6 x-4)-2 x^3(x^2-3)$

Let's dive into our example problem: $3 x^2(5 x^3+6 x-4)-2 x^3(x^2-3)$. Our goal is to simplify this expression by performing the indicated operations, which primarily involve distribution and combining like terms. This type of problem is fundamental in algebra and often appears in higher-level math courses. Mastering the steps involved in simplifying polynomial expressions is essential for success in mathematics. The expression itself contains two main parts, each requiring the distributive property to eliminate the parentheses. We'll then identify and combine like terms to arrive at the simplest form. Remember, the order of operations (PEMDAS/BODMAS) is crucial here: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). By following these steps carefully, we'll transform the complex expression into a more manageable and understandable form. The ability to simplify polynomial expressions is not just a mathematical skill; it's a valuable problem-solving tool that can be applied in various contexts.

Step 1: Distribute $3x^2$ into $(5x^3 + 6x - 4)$

The first step in simplifying the expression is to distribute the $3x^2$ term into the parentheses $(5x^3 + 6x - 4)$. This means we need to multiply $3x^2$ by each term inside the parentheses individually. Remember the rule of exponents: when multiplying terms with the same base, you add the exponents. Distributing correctly is crucial for simplifying polynomial expressions, as it lays the foundation for combining like terms later on. So, let's break it down:

• $3x^2 * 5x^3 = (3 * 5) * (x^2 * x^3) = 15x^{2+3} = 15x^5$
• $3x^2 * 6x = (3 * 6) * (x^2 * x^1) = 18x^{2+1} = 18x^3$
• $3x^2 * -4 = (3 * -4) * x^2 = -12x^2$

Putting these together, we get: $15x^5 + 18x^3 - 12x^2$. This is the result of the first distribution. Careful attention to the rules of exponents and multiplication ensures accuracy in this step. By distributing the term, we've effectively removed the parentheses and created individual terms that can be further simplified. This process is a cornerstone of algebraic manipulation and is used extensively in solving equations and simplifying expressions. Understanding this step thoroughly will make the subsequent steps much easier to handle. The distributed form allows us to see the individual components more clearly, making the next step of combining like terms more straightforward.

Step 2: Distribute $-2x^3$ into $(x^2 - 3)$

Next, we need to distribute the $-2x^3$ term into the parentheses $(x^2 - 3)$. Similar to the previous step, we multiply $-2x^3$ by each term inside the parentheses, paying close attention to the signs and exponents. Accurate distribution of the negative term is essential to avoid errors in the final simplified expression. Remember that multiplying a negative term by a positive term results in a negative term, and multiplying a negative term by a negative term results in a positive term. Let's break this down:

• $-2x^3 * x^2 = -2 * (x^3 * x^2) = -2x^{3+2} = -2x^5$
• $-2x^3 * -3 = (-2 * -3) * x^3 = 6x^3$

Combining these, we have: $-2x^5 + 6x^3$. This is the result of the second distribution. The correct application of the distributive property and the rules of exponents is paramount in this step. The negative sign in front of $2x^3$ plays a crucial role, as it affects the signs of the resulting terms. By distributing this term, we've further expanded the expression and prepared it for the final simplification step: combining like terms. This step highlights the importance of precision in algebraic manipulations. A small error in the sign or exponent can lead to a completely different final result. Therefore, double-checking each multiplication and exponent addition is a good practice.

Step 3: Combine Like Terms

Now that we've distributed both terms, our expression looks like this: $15x^5 + 18x^3 - 12x^2 - 2x^5 + 6x^3$. The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. This is a crucial step in simplifying polynomial expressions, as it reduces the number of terms and makes the expression more manageable. Identifying like terms involves looking for terms with identical variable and exponent combinations. In our expression, we have:

• $15x^5$ and $-2x^5$ are like terms.
• $18x^3$ and $6x^3$ are like terms.
• $-12x^2$ has no like terms in the expression.

To combine like terms, we simply add or subtract their coefficients (the numerical part of the term). So, let's combine them:

• $15x^5 - 2x^5 = (15 - 2)x^5 = 13x^5$
• $18x^3 + 6x^3 = (18 + 6)x^3 = 24x^3$

And $-12x^2$ remains as it is since there are no other terms with $x^2$. Correctly identifying and combining like terms is essential for arriving at the simplest form of the expression. Misidentifying terms or incorrectly adding coefficients can lead to errors. This step is often where mistakes occur, so it's important to be meticulous. By combining like terms, we're essentially consolidating the expression into its most compact form. This makes it easier to work with in subsequent calculations or analyses. The final simplified expression will have fewer terms and be easier to understand and interpret.

Step 4: Write the Simplified Expression

After combining like terms, we have $13x^5 + 24x^3 - 12x^2$. This is the simplified form of the original expression. Writing the simplified expression in the correct order, typically from the highest power of the variable to the lowest, enhances clarity and readability. This is a conventional practice in mathematics, making it easier for others to understand and work with the expression. So, the final simplified expression is:

$13x^5 + 24x^3 - 12x^2$

This is our final answer. We started with a more complex expression and, through the steps of distribution and combining like terms, we've arrived at a simplified version. The ability to simplify expressions like this is a fundamental skill in algebra and is essential for solving more complex problems. This simplified expression is easier to work with in further calculations, such as solving equations or evaluating the expression for specific values of x. The process of simplification not only makes the expression more concise but also reveals its underlying structure more clearly. This final step demonstrates the power of algebraic manipulation in reducing complex expressions to their most basic forms. It showcases the importance of following a systematic approach to problem-solving in mathematics.

Conclusion

In this guide, we've walked through the process of simplifying a polynomial expression, $3 x^2(5 x^3+6 x-4)-2 x^3(x^2-3)$, step by step. We covered the importance of distribution, combining like terms, and writing the final expression in a simplified form. Mastering these techniques will not only help you in algebra but also in various other fields of mathematics and science. Understanding how to manipulate and simplify polynomial expressions is a crucial skill for anyone working with mathematical concepts. The ability to break down a complex expression into simpler components allows for a clearer understanding and easier manipulation. Remember, practice makes perfect. The more you work with these types of problems, the more comfortable and confident you'll become in your ability to solve them. Don't hesitate to review the steps and try similar problems on your own. The key is to understand the underlying principles and apply them systematically.

For further learning and practice on polynomial operations, you can explore resources like Khan Academy's Algebra I section.