Simplifying Polynomial Expressions A Step-by-Step Guide

In the realm of mathematics, polynomial expressions are fundamental building blocks. These expressions, consisting of variables and coefficients, often appear complex at first glance. However, with a systematic approach, simplifying them becomes a manageable task. This article delves into the process of simplifying the polynomial expression $3x(-2x + 7) - 5(x - 1)(4x - 3)$, providing a detailed, step-by-step guide to arrive at the correct solution. Mastering this process is crucial for success in algebra and beyond, as it lays the groundwork for more advanced mathematical concepts.

The journey of simplifying polynomial expressions begins with a clear understanding of the order of operations and the distributive property. These are the cornerstones upon which we build our simplification strategy. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which we perform operations. The distributive property, on the other hand, allows us to multiply a term across a sum or difference within parentheses. By carefully applying these principles, we can systematically unravel the complexities of polynomial expressions and arrive at their simplest forms.

The expression we aim to simplify, $3x(-2x + 7) - 5(x - 1)(4x - 3)$, presents a blend of multiplication and subtraction, involving both single-term and multi-term polynomials. To tackle this, we'll first focus on the multiplications, carefully distributing terms and combining like terms as we proceed. The initial step involves distributing $3x$ across the terms within the first parenthesis, and then addressing the product of the two binomials in the second term. This strategic breakdown allows us to manage the complexity and minimize the risk of errors. As we move through each step, we'll emphasize clarity and precision, ensuring that each operation is performed accurately and logically. This meticulous approach not only leads to the correct answer but also fosters a deeper understanding of the underlying mathematical principles.

To effectively simplify the given polynomial expression, $3x(-2x + 7) - 5(x - 1)(4x - 3)$, we will methodically work through each step. First, we'll apply the distributive property to remove the parentheses, then we'll combine like terms to arrive at the simplified form. This process requires careful attention to detail and a solid understanding of algebraic principles.

Step 1: Distribute $3x$ in the first term

Our initial focus is on the term $3x(-2x + 7)$. We apply the distributive property, multiplying $3x$ by each term inside the parentheses:

$3x * (-2x) = -6x^2$

$3x * 7 = 21x$

Combining these results, we get $-6x^2 + 21x$. This step demonstrates the power of the distributive property in expanding expressions and setting the stage for further simplification. By breaking down the expression into smaller, manageable parts, we minimize the chance of errors and maintain a clear path towards the solution. The careful application of this property is crucial in simplifying polynomial expressions, as it allows us to transform complex expressions into more manageable forms.

Step 2: Multiply the binomials $(x - 1)(4x - 3)$

Next, we address the binomial multiplication $(x - 1)(4x - 3)$. We use the FOIL method (First, Outer, Inner, Last) to ensure we multiply each term in the first binomial by each term in the second binomial:

• First: $x * 4x = 4x^2$
• Outer: $x * (-3) = -3x$
• Inner: $-1 * 4x = -4x$
• Last: $-1 * (-3) = 3$

Combining these terms, we get $4x^2 - 3x - 4x + 3$. Now, we combine the like terms $-3x$ and $-4x$ to get $-7x$. So, the result of the binomial multiplication is $4x^2 - 7x + 3$. This step highlights the importance of systematic multiplication when dealing with binomials. The FOIL method provides a structured approach, ensuring that no term is missed and that the resulting expression is accurate. Mastering binomial multiplication is essential for simplifying more complex polynomial expressions and solving algebraic equations.

Step 3: Distribute $-5$ in the second term

Now we need to distribute the $-5$ across the result from Step 2, which is $(4x^2 - 7x + 3)$:

$-5 * (4x^2) = -20x^2$

$-5 * (-7x) = 35x$

$-5 * 3 = -15$

Combining these, we get $-20x^2 + 35x - 15$. This step demonstrates the importance of paying attention to signs when distributing. The negative sign in front of the 5 affects the sign of each term inside the parentheses, and careful application of the distributive property is crucial to avoid errors. This step further simplifies the expression, bringing us closer to the final solution. The ability to accurately distribute terms, especially with negative coefficients, is a fundamental skill in algebra and is essential for simplifying polynomial expressions.

Step 4: Combine all terms

Finally, we combine the results from Step 1 and Step 3:

$(-6x^2 + 21x) + (-20x^2 + 35x - 15)$

Combine the $x^2$ terms: $-6x^2 - 20x^2 = -26x^2$

Combine the $x$ terms: $21x + 35x = 56x$

The constant term is $-15$.

Combining these simplified terms, we arrive at the final simplified polynomial expression: $-26x^2 + 56x - 15$. This final step underscores the importance of identifying and combining like terms. By grouping terms with the same variable and exponent, we can reduce the expression to its simplest form. This process not only provides the final answer but also demonstrates the underlying structure of the polynomial. The ability to accurately combine like terms is a fundamental skill in algebra and is essential for simplifying and solving polynomial expressions.

By following these steps, we have successfully simplified the polynomial expression $3x(-2x + 7) - 5(x - 1)(4x - 3)$. The final simplified form is:

-26x^2 + 56x - 15

Therefore, the correct answer is A. $-26x^2 + 56x - 15$.

In conclusion, simplifying polynomial expressions requires a systematic approach, careful application of the distributive property, and accurate combination of like terms. Mastering these techniques is essential for success in algebra and higher-level mathematics. By understanding the underlying principles and practicing regularly, you can confidently tackle even the most complex polynomial expressions. This step-by-step guide has provided a framework for simplifying polynomials, and with continued practice, you can develop the skills necessary to excel in this area of mathematics.

• Polynomial expressions
• Simplifying polynomials
• Distributive property
• Combining like terms
• FOIL method
• Algebra
• Mathematics
• Step-by-step guide
• Polynomial simplification
• Mathematical expressions