Simplifying (x-4)(x^2+3x-5) Step-by-Step Solution

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Introduction

In the realm of mathematics, simplifying expressions is a fundamental skill that paves the way for tackling more complex problems. This article delves into the process of simplifying the expression (x-4)(x^2+3x-5), a common task encountered in algebra. We will explore the step-by-step methodology to arrive at the simplest form, offering a clear understanding of the underlying concepts and techniques. The goal is not just to provide the answer but to equip you with the knowledge to confidently approach similar problems. Whether you are a student grappling with algebraic expressions or someone looking to refresh your mathematical skills, this guide will serve as a valuable resource.

Understanding the Problem

Before diving into the solution, it's crucial to understand the nature of the problem. We are presented with a product of two polynomials: a binomial (x-4) and a trinomial (x^2+3x-5). The task is to expand this product and then combine like terms to obtain the simplest form of the expression. This involves applying the distributive property, a core concept in algebra, which dictates how to multiply a sum by a term. We need to meticulously multiply each term in the binomial with each term in the trinomial, ensuring we account for all combinations. This process requires careful attention to signs and exponents to avoid common errors. Once we have expanded the product, we will identify and combine like terms, which are terms with the same variable raised to the same power. This step is crucial for simplifying the expression to its most concise form. By understanding the problem thoroughly, we set the stage for a successful and accurate solution.

Step-by-Step Solution

To simplify the expression (x-4)(x^2+3x-5), we will follow a step-by-step approach that leverages the distributive property. This method ensures that we systematically multiply each term of the first polynomial by each term of the second polynomial.

Step 1: Distribute the First Term

We begin by distributing the first term of the binomial, which is x, across the trinomial:

x(x^2 + 3x - 5) = x^3 + 3x^2 - 5x*

This step involves multiplying x by each term inside the parentheses. When multiplying terms with exponents, we add the exponents together (e.g., x * x^2 = x^(1+2) = x^3). It's crucial to pay close attention to the signs of the terms as well. Here, all terms are positive, so the products remain positive.

Step 2: Distribute the Second Term

Next, we distribute the second term of the binomial, which is -4, across the trinomial:

-4(x^2 + 3x - 5) = -4x^2 - 12x + 20

In this step, we multiply -4 by each term inside the parentheses. Notice how the negative sign affects the signs of the resulting terms. For example, -4 multiplied by 3x results in -12x, and -4 multiplied by -5 results in +20. Careful attention to these sign changes is essential for accurate calculations.

Step 3: Combine the Results

Now, we combine the results from Step 1 and Step 2:

(x^3 + 3x^2 - 5x) + (-4x^2 - 12x + 20)

This step involves writing down all the terms we obtained from the distribution steps. We are essentially adding the two expanded expressions together. The next step will involve identifying and combining like terms to simplify this expression further.

Step 4: Combine Like Terms

Like terms are terms that have the same variable raised to the same power. In our expression, we can identify the following like terms:

  • x^3: There is only one x^3 term.
  • x^2: We have 3x^2 and -4x^2.
  • x: We have -5x and -12x.
  • Constants: We have +20.

Combining these like terms, we get:

x^3 + (3x^2 - 4x^2) + (-5x - 12x) + 20 = x^3 - x^2 - 17x + 20

This step is where the expression is truly simplified. By adding or subtracting the coefficients of like terms, we reduce the number of terms in the expression, making it more concise and easier to work with.

Step 5: Final Simplified Form

The simplest form of the expression (x-4)(x^2+3x-5) is:

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

This is the final answer. We have successfully expanded the original expression and combined like terms to arrive at its most simplified form. This form is equivalent to the original expression but is much easier to understand and use in further calculations.

Analyzing the Answer Choices

Now that we have derived the simplest form of the expression, let's analyze the answer choices provided to identify the correct one. The choices typically present variations of the expanded expression, with common errors arising from incorrect distribution, sign mistakes, or misidentification of like terms. By comparing our derived answer with the choices, we can not only confirm the correct solution but also understand the potential pitfalls in the simplification process. This analytical step reinforces the importance of meticulousness and attention to detail in algebraic manipulations. Understanding why certain choices are incorrect is as crucial as identifying the correct answer, as it helps in avoiding similar errors in future problems.

The answer choices provided are:

  • A. x^3 - x^2 - 6x - 20
  • B. x^3 + 7x^2 - 17x + 20
  • C. x^3 - x^2 + 7x - 20
  • D. x^3 - x^2 - 17x + 20

Comparing our simplified form (x^3 - x^2 - 17x + 20) with the answer choices, we can clearly see that option D matches our result. The other options contain errors in the coefficients of the terms, highlighting the importance of careful calculation and sign management during the simplification process. For instance, option A has an incorrect coefficient for the x term, while options B and C have errors in the x^2 and x terms, respectively. This comparison underscores the significance of each step in the simplification process and the need for thoroughness in algebraic manipulations.

Common Mistakes to Avoid

When simplifying algebraic expressions like (x-4)(x^2+3x-5), several common mistakes can lead to incorrect answers. Being aware of these pitfalls and actively avoiding them is crucial for achieving accuracy. One of the most frequent errors is incorrect distribution. This can occur when students forget to multiply each term in the first polynomial by every term in the second polynomial. For example, they might multiply x by x^2 and 3x but forget to multiply it by -5. To avoid this, it's helpful to use a systematic approach, ensuring each term is accounted for. Another common mistake is sign errors. When distributing a negative term, such as -4 in our expression, it's essential to correctly apply the negative sign to each term inside the parentheses. Forgetting to do so can lead to incorrect signs in the expanded expression. Misidentifying like terms is another potential pitfall. Like terms must have the same variable raised to the same power. Students sometimes mistakenly combine terms with different powers, leading to an incorrect simplified form. Finally, arithmetic errors in addition and subtraction can also lead to mistakes. Even if the distribution and identification of like terms are correct, a simple addition or subtraction error can result in the wrong answer. To minimize these errors, it's advisable to double-check each step and use techniques like writing out each step clearly to reduce the chances of oversight. By being mindful of these common mistakes and employing strategies to avoid them, students can significantly improve their accuracy in simplifying algebraic expressions.

Practice Problems

To solidify your understanding of simplifying expressions, working through practice problems is essential. Here are a few problems similar to the one we just solved. Attempting these will help you hone your skills and identify any areas where you might need further review. Remember to follow the step-by-step approach we outlined earlier: distribute, combine like terms, and double-check your work. These exercises will not only improve your procedural fluency but also deepen your conceptual understanding of algebraic manipulation.

  1. Simplify: (2x + 1)(x^2 - 4x + 3)
  2. Simplify: (x - 3)(x^2 + 2x - 1)
  3. Simplify: (3x - 2)(2x^2 - x + 4)

These problems offer a range of scenarios, including different coefficients and signs, to challenge your understanding. After attempting these problems, it's beneficial to compare your solutions with the correct answers to identify any discrepancies and understand where errors might have occurred. Consistent practice is key to mastering the art of simplifying algebraic expressions and building confidence in your mathematical abilities.

Conclusion

In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics that requires a systematic approach and attention to detail. Through this article, we have explored the process of simplifying the expression (x-4)(x^2+3x-5), emphasizing the importance of the distributive property, combining like terms, and avoiding common mistakes. We broke down the solution into manageable steps, providing a clear and concise methodology that can be applied to similar problems. By analyzing the answer choices, we reinforced the correct solution and highlighted potential pitfalls in the simplification process. Furthermore, we identified common mistakes to avoid and provided practice problems to solidify your understanding and skills. Mastering this skill not only enhances your algebraic proficiency but also builds a strong foundation for tackling more advanced mathematical concepts. Remember, consistent practice and a methodical approach are key to success in simplifying expressions and achieving mathematical fluency. With the knowledge and techniques presented in this guide, you are well-equipped to confidently approach and solve a wide range of algebraic problems.