How To Simplify (2x - 3)(5x^4 - 7x^3 + 6x^2 - 9) Polynomial Expression

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Polynomial expressions, a fundamental concept in algebra, often appear complex and daunting at first glance. However, with a systematic approach and a clear understanding of the distributive property, simplifying these expressions can become a manageable task. In this comprehensive guide, we will break down the process of simplifying the polynomial expression (2x−3)(5x4−7x3+6x2−9)(2x - 3)(5x^4 - 7x^3 + 6x^2 - 9), providing a step-by-step solution and insightful explanations to enhance your understanding of polynomial manipulation. Mastering the simplification of polynomial expressions is crucial for success in various mathematical disciplines, including calculus, linear algebra, and differential equations. It also forms the bedrock for solving real-world problems in fields such as physics, engineering, and economics, where mathematical models often involve polynomial relationships. By diligently working through the steps outlined in this guide, you will gain the confidence and skills necessary to tackle more complex polynomial expressions and apply your knowledge to diverse problem-solving scenarios. This process not only simplifies the given expression but also reinforces key algebraic concepts, making it a valuable exercise for students and professionals alike. Through practice and application, you can transform the perceived complexity of polynomial expressions into a clear and understandable mathematical landscape.

Understanding the Distributive Property

The cornerstone of simplifying polynomial expressions lies in the distributive property. This property, a fundamental principle in algebra, dictates how to multiply a single term by an expression enclosed in parentheses. In essence, the distributive property states that for any numbers a, b, and c, the following holds true: a(b + c) = ab + ac. This simple yet powerful rule forms the basis for expanding and simplifying polynomial expressions. When dealing with more complex expressions involving multiple terms, the distributive property is applied repeatedly to ensure that each term within the first set of parentheses is multiplied by every term within the second set of parentheses. Consider the expression (x + 2)(x + 3). To simplify this, we distribute the 'x' from the first parentheses to both terms in the second parentheses, resulting in x(x + 3) = x^2 + 3x. Similarly, we distribute the '2' from the first parentheses to both terms in the second parentheses, yielding 2(x + 3) = 2x + 6. Finally, we combine the resulting terms: x^2 + 3x + 2x + 6. By combining like terms (3x and 2x), we arrive at the simplified expression: x^2 + 5x + 6. This example illustrates the core mechanics of the distributive property in action. In the context of the given expression, (2x−3)(5x4−7x3+6x2−9)(2x - 3)(5x^4 - 7x^3 + 6x^2 - 9), we will apply the distributive property in a similar manner, ensuring that each term in the first binomial (2x - 3) is multiplied by each term in the second polynomial (5x4−7x3+6x2−9)(5x^4 - 7x^3 + 6x^2 - 9). This meticulous distribution is the key to unraveling the expression and arriving at its simplified form.

Step-by-Step Solution

Let's embark on a step-by-step journey to simplify the polynomial expression (2x−3)(5x4−7x3+6x2−9)(2x - 3)(5x^4 - 7x^3 + 6x^2 - 9). This process will involve a careful application of the distributive property, ensuring that each term is multiplied correctly and like terms are combined effectively.

Step 1: Distribute the First Term

The first step involves distributing the term '2x' from the first binomial (2x - 3) across all the terms in the second polynomial (5x4−7x3+6x2−9)(5x^4 - 7x^3 + 6x^2 - 9). This means we will multiply 2x by each term within the polynomial individually. Let's break it down:

  • 2x * 5x^4 = 10x^5
  • 2x * (-7x^3) = -14x^4
  • 2x * 6x^2 = 12x^3
  • 2x * (-9) = -18x

Combining these results, we get the expression: 10x^5 - 14x^4 + 12x^3 - 18x. This represents the partial expansion resulting from distributing the first term of the binomial.

Step 2: Distribute the Second Term

Next, we distribute the second term '-3' from the first binomial (2x - 3) across all the terms in the second polynomial (5x4−7x3+6x2−9)(5x^4 - 7x^3 + 6x^2 - 9). Again, we multiply -3 by each term within the polynomial:

  • -3 * 5x^4 = -15x^4
  • -3 * (-7x^3) = 21x^3
  • -3 * 6x^2 = -18x^2
  • -3 * (-9) = 27

Combining these results, we obtain the expression: -15x^4 + 21x^3 - 18x^2 + 27. This represents the partial expansion resulting from distributing the second term of the binomial.

Step 3: Combine Like Terms

Now, we combine the two partial expansions obtained in the previous steps. This involves adding the expressions: (10x^5 - 14x^4 + 12x^3 - 18x) + (-15x^4 + 21x^3 - 18x^2 + 27). To do this effectively, we group like terms together:

  • Terms with x^5: 10x^5 (only one term)
  • Terms with x^4: -14x^4 - 15x^4 = -29x^4
  • Terms with x^3: 12x^3 + 21x^3 = 33x^3
  • Terms with x^2: -18x^2 (only one term)
  • Terms with x: -18x (only one term)
  • Constant terms: 27 (only one term)

Step 4: Write the Simplified Expression

Finally, we combine the results from the previous step to write the simplified expression. This involves arranging the terms in descending order of their exponents:

10x^5 - 29x^4 + 33x^3 - 18x^2 - 18x + 27

Therefore, the simplified form of the given polynomial expression (2x−3)(5x4−7x3+6x2−9)(2x - 3)(5x^4 - 7x^3 + 6x^2 - 9) is 10x^5 - 29x^4 + 33x^3 - 18x^2 - 18x + 27. This matches option D in the given choices.

Analyzing the Options

Having meticulously simplified the polynomial expression, let's analyze the provided options to confirm our result and understand why the other options are incorrect. This critical step reinforces the importance of accuracy in polynomial manipulation and helps in identifying common errors.

  • Option A: 10x^5 - x^4 - 9x^3 - 18x^2 - 18x - 27

    This option is incorrect. The coefficients of the x^4 and x^3 terms are significantly different from our calculated result. The constant term also has the wrong sign. These discrepancies indicate errors in the distribution or combination of like terms.

  • Option B: 10x^5 + x^4 + 33x^3 + 18x^2 + 18x + 27

    This option is also incorrect. The signs of several terms, including x^4, x^2, and x, are incorrect. This suggests potential sign errors during the distribution process, where negative terms were not handled properly.

  • Option C: 10x^5 + 29x^4 - 33x^3 + 18x^2 + 18x - 27

    This option presents incorrect signs for the x^4 and x^3 terms. The error likely stems from miscalculations during the combination of like terms, where the signs were not properly accounted for.

  • Option D: 10x^5 - 29x^4 + 33x^3 - 18x^2 - 18x + 27

    This option perfectly matches our simplified expression. Each term, along with its coefficient and sign, aligns with the result obtained through our step-by-step solution. This confirms the accuracy of our calculations and the validity of option D.

By systematically comparing our result with each option, we not only validate our solution but also gain a deeper understanding of the common mistakes that can occur during polynomial simplification. This analytical approach is crucial for building confidence and proficiency in algebraic manipulations.

Common Mistakes to Avoid

Simplifying polynomial expressions, while systematic, is prone to errors if careful attention is not paid to detail. Understanding the common pitfalls can significantly improve accuracy and efficiency. Here are some frequent mistakes to watch out for:

  1. Sign Errors: This is perhaps the most common mistake. When distributing a negative term, ensure that the sign of every term within the parentheses is correctly flipped. For example, in the expression -3(2x - 5), it's crucial to distribute the negative sign to both terms, resulting in -6x + 15, not -6x - 15.

  2. Incorrect Multiplication of Exponents: When multiplying terms with exponents, remember the rule: x^m * x^n = x^(m+n). A common mistake is to multiply the exponents instead of adding them. For instance, 2x^2 * 3x^3 should be 6x^5, not 6x^6.

  3. Forgetting to Distribute: Ensure that every term in the first set of parentheses is multiplied by every term in the second set. A partial distribution will lead to an incorrect simplification. Double-check your work to confirm that all terms have been accounted for.

  4. Combining Unlike Terms: Only like terms (terms with the same variable and exponent) can be combined. Avoid adding or subtracting terms with different variables or exponents. For example, 3x^2 and 5x cannot be combined.

  5. Order of Operations: Adhere to the order of operations (PEMDAS/BODMAS). Exponents should be handled before multiplication and division, and addition and subtraction should be performed last. A deviation from this order can lead to incorrect results.

  6. Careless Arithmetic: Simple arithmetic errors, such as incorrect addition or subtraction of coefficients, can derail the entire simplification process. Take your time and double-check your calculations.

By being mindful of these common mistakes and implementing strategies to avoid them, you can significantly enhance your accuracy and confidence in simplifying polynomial expressions. Consistent practice and a methodical approach are key to mastering this essential algebraic skill.

Practice Problems

To solidify your understanding and hone your skills in simplifying polynomial expressions, working through practice problems is essential. These problems provide an opportunity to apply the concepts and techniques discussed in this guide and identify areas where further practice may be needed. Here are a few practice problems, ranging in complexity, to challenge your abilities:

  1. Simplify: (x + 4)(x - 2)
  2. Simplify: (3x - 1)(2x + 5)
  3. Simplify: (x^2 + 2x - 3)(x + 1)
  4. Simplify: (2x - 3)(x^2 - 4x + 5)
  5. Simplify: (x + 2)(x - 2)(x + 3)

For each problem, follow the step-by-step approach outlined in this guide: distribute the terms carefully, combine like terms accurately, and write the simplified expression in descending order of exponents. After attempting the problems, compare your solutions with the answers provided below to assess your progress.

  • Solution 1: x^2 + 2x - 8
  • Solution 2: 6x^2 + 13x - 5
  • Solution 3: x^3 + 3x^2 - x - 3
  • Solution 4: 2x^3 - 11x^2 + 22x - 15
  • Solution 5: x^3 + 3x^2 - 4x - 12

If your solutions match the answers, congratulations! You have a strong grasp of the concepts. If you encountered any difficulties, revisit the relevant sections of this guide, review the steps involved, and try the problem again. Consistent practice and a willingness to learn from mistakes are crucial for mastering polynomial simplification. Consider seeking additional practice problems from textbooks, online resources, or your instructor to further strengthen your skills. Remember, the more you practice, the more confident and proficient you will become in manipulating polynomial expressions.

Conclusion

In conclusion, simplifying polynomial expressions is a fundamental skill in algebra that requires a systematic approach and a strong understanding of the distributive property. By meticulously distributing terms, combining like terms, and paying close attention to signs and exponents, you can effectively simplify even complex expressions. This guide has provided a step-by-step solution to the expression (2x−3)(5x4−7x3+6x2−9)(2x - 3)(5x^4 - 7x^3 + 6x^2 - 9), highlighting the key steps involved in the simplification process. We have also analyzed common mistakes to avoid and provided practice problems to solidify your understanding. Mastering polynomial simplification not only enhances your algebraic skills but also lays a solid foundation for more advanced mathematical concepts. Whether you are a student learning algebra for the first time or a professional applying mathematical models in your field, the ability to simplify polynomial expressions is an invaluable asset. Remember, consistent practice and a methodical approach are the keys to success. Embrace the challenge, work through the problems diligently, and you will find that simplifying polynomial expressions becomes a clear and manageable task.