Simplifying Polynomial Expressions A Step By Step Guide

Polynomial expressions are fundamental in algebra, and simplifying them is a crucial skill for various mathematical operations. This article aims to provide a comprehensive, step-by-step guide on how to simplify polynomial expressions, focusing on the expression: $(5x^3 + 4x^2 + 6x - 3) + (-2x^3 + 3x^2 - 7x + 9) - (5x + 3)(4x^2 + x - 1)$. We will break down the process into manageable steps, ensuring clarity and understanding for readers of all levels. By mastering these techniques, you'll be well-equipped to tackle more complex algebraic problems.

Before diving into the simplification process, it's essential to understand the basics of polynomial expressions. A polynomial expression is a mathematical expression consisting of variables (usually denoted by letters like 'x') and coefficients, combined using addition, subtraction, and multiplication. The exponents of the variables must be non-negative integers. For example, $5x^3 + 4x^2 + 6x - 3$ is a polynomial expression, while $5x^{-2}$ or $4 √x$ are not, due to the negative and fractional exponents, respectively.

Polynomials are classified by their degree, which is the highest power of the variable in the expression. In our example, $5x^3 + 4x^2 + 6x - 3$, the highest power of 'x' is 3, making it a third-degree polynomial, also known as a cubic polynomial. The terms in a polynomial are the individual components separated by addition or subtraction. In the given expression, the terms are $5x^3$, $4x^2$, $6x$, and -3. Understanding these basic concepts will help you approach the simplification process with confidence.

When simplifying polynomial expressions, the key is to combine like terms. Like terms are terms that have the same variable raised to the same power. For instance, $5x^3$ and $-2x^3$ are like terms because they both have $x^3$. Similarly, $4x^2$ and $3x^2$ are like terms. Terms like $6x$ and $-7x$ are also like terms. Constant terms, such as -3 and 9, are also considered like terms. The process of simplification involves identifying and combining these like terms by adding or subtracting their coefficients. This not only makes the expression more concise but also easier to work with in subsequent calculations or problem-solving scenarios.

In our given expression, $(5x^3 + 4x^2 + 6x - 3) + (-2x^3 + 3x^2 - 7x + 9) - (5x + 3)(4x^2 + x - 1)$, the first step in simplifying is to expand the product $(5x + 3)(4x^2 + x - 1)$. This involves using the distributive property, which states that $a(b + c) = ab + ac$. We need to multiply each term in the first parenthesis by each term in the second parenthesis.

To expand $(5x + 3)(4x^2 + x - 1)$, we multiply $5x$ by each term in the second parenthesis and then multiply 3 by each term in the second parenthesis. This can be written as:

$5x(4x^2 + x - 1) + 3(4x^2 + x - 1)$

Now, let's perform the multiplication:

$5x * 4x^2 = 20x^3$ $5x * x = 5x^2$ $5x * -1 = -5x$

And,

$3 * 4x^2 = 12x^2$ $3 * x = 3x$ $3 * -1 = -3$

Combining these terms, we get:

$20x^3 + 5x^2 - 5x + 12x^2 + 3x - 3$

Next, we combine like terms within this expanded expression. The like terms are $5x^2$ and $12x^2$, and $-5x$ and $3x$. Adding the coefficients, we have:

$5x^2 + 12x^2 = 17x^2$ $-5x + 3x = -2x$

So, the expanded and simplified form of $(5x + 3)(4x^2 + x - 1)$ is:

$20x^3 + 17x^2 - 2x - 3$

This result is crucial for the next step in simplifying the entire polynomial expression. Expanding the product correctly ensures that we can accurately combine like terms in the subsequent steps, leading to the correct simplified form. By breaking down the multiplication and combining like terms within the expanded product, we make the overall simplification process more manageable and less prone to errors.

Now that we have expanded the product, we can substitute it back into the original expression and remove the parentheses. The original expression is $(5x^3 + 4x^2 + 6x - 3) + (-2x^3 + 3x^2 - 7x + 9) - (5x + 3)(4x^2 + x - 1)$. Replacing the expanded product, we get:

$(5x^3 + 4x^2 + 6x - 3) + (-2x^3 + 3x^2 - 7x + 9) - (20x^3 + 17x^2 - 2x - 3)$

The next step is to remove the parentheses. When removing parentheses preceded by a positive sign, the signs of the terms inside the parentheses remain the same. When the parentheses are preceded by a negative sign, the signs of the terms inside the parentheses are reversed. Applying this rule, we have:

$5x^3 + 4x^2 + 6x - 3 - 2x^3 + 3x^2 - 7x + 9 - 20x^3 - 17x^2 + 2x + 3$

Now, we need to combine like terms. This involves grouping terms with the same variable and exponent together and then adding or subtracting their coefficients. Let's group the like terms:

$(5x^3 - 2x^3 - 20x^3) + (4x^2 + 3x^2 - 17x^2) + (6x - 7x + 2x) + (-3 + 9 + 3)$

Next, we add or subtract the coefficients:

For the $x^3$ terms: $5x^3 - 2x^3 - 20x^3 = (5 - 2 - 20)x^3 = -17x^3$ For the $x^2$ terms: $4x^2 + 3x^2 - 17x^2 = (4 + 3 - 17)x^2 = -10x^2$ For the $x$ terms: $6x - 7x + 2x = (6 - 7 + 2)x = 1x = x$ For the constant terms: $-3 + 9 + 3 = 9$

Combining these results, we get the simplified polynomial expression:

$-17x^3 - 10x^2 + x + 9$

This step is critical in reducing the complex expression to its simplest form. By carefully removing the parentheses and accurately combining like terms, we ensure that the final expression is both concise and correct. The process of grouping like terms and then performing the arithmetic on their coefficients is a fundamental technique in polynomial simplification.

After performing the necessary steps, we have arrived at the final simplified expression. In Step 1, we expanded the product $(5x + 3)(4x^2 + x - 1)$ to get $20x^3 + 17x^2 - 2x - 3$. In Step 2, we removed parentheses and combined like terms from the original expression, resulting in $-17x^3 - 10x^2 + x + 9$. Therefore, the simplified form of the given polynomial expression is:

$-17x^3 - 10x^2 + x + 9$

This result matches option A in the multiple-choice answers provided. The simplification process involved careful application of the distributive property, accurate arithmetic, and a systematic approach to combining like terms. Each step was crucial in arriving at the correct answer. A mistake in any of these steps could lead to an incorrect simplification. Polynomial simplification is a core concept in algebra, and mastering it is essential for success in more advanced mathematical topics.

In conclusion, simplifying polynomial expressions is a fundamental skill in algebra that requires a step-by-step approach. By expanding products, removing parentheses, and combining like terms, we can reduce complex expressions to their simplest form. The example we worked through, $(5x^3 + 4x^2 + 6x - 3) + (-2x^3 + 3x^2 - 7x + 9) - (5x + 3)(4x^2 + x - 1)$, illustrates this process effectively. The key takeaways include the importance of the distributive property, the rules for removing parentheses, and the technique of combining like terms. Practicing these steps with various polynomial expressions will solidify your understanding and improve your ability to solve algebraic problems. The simplified expression, $-17x^3 - 10x^2 + x + 9$, demonstrates the power of these techniques in making complex expressions more manageable and understandable.

By following this guide, you should now have a clear understanding of how to simplify polynomial expressions. Remember to break down complex problems into smaller, manageable steps, and always double-check your work to ensure accuracy. With practice, you'll become proficient in simplifying polynomial expressions and applying this skill to solve a wide range of mathematical problems.