Find The Equivalent Expression Of (d - 7)(8d² - 5)
Understanding algebraic expressions and their equivalent forms is a fundamental concept in mathematics. In this article, we will delve into the process of expanding and simplifying polynomial expressions, focusing on the specific expression (d - 7)(8d² - 5). We aim to identify the equivalent form among the provided options. This exploration will not only reinforce your understanding of polynomial multiplication but also enhance your ability to manipulate algebraic expressions effectively.
Expanding Polynomial Expressions
In the realm of algebra, expanding polynomial expressions involves multiplying terms across parentheses and combining like terms to arrive at a simplified form. This process is crucial for solving equations, simplifying complex expressions, and gaining a deeper understanding of algebraic relationships. When dealing with expressions such as (d - 7)(8d² - 5), the distributive property becomes our primary tool. This property dictates that each term within the first set of parentheses must be multiplied by each term within the second set of parentheses.
To illustrate, let's break down the expansion of (d - 7)(8d² - 5) step by step. First, we multiply the term 'd' from the first parentheses by both terms in the second parentheses:
- d * 8d² = 8d³
- d * -5 = -5d
Next, we multiply the term '-7' from the first parentheses by both terms in the second parentheses:
- -7 * 8d² = -56d²
- -7 * -5 = 35
Now, we combine all the resulting terms:
8d³ - 5d - 56d² + 35
To present the expression in standard form, we arrange the terms in descending order of their exponents:
8d³ - 56d² - 5d + 35
This systematic approach ensures that we account for every term and its corresponding product, minimizing the risk of errors. By mastering the art of expanding polynomial expressions, you'll be well-equipped to tackle more complex algebraic problems with confidence.
Step-by-Step Solution
Let's meticulously walk through the process of expanding the expression (d - 7)(8d² - 5). This detailed step-by-step solution will not only provide the answer but also reinforce the methodology for handling such algebraic manipulations. Our primary tool in this endeavor is the distributive property, which, as mentioned earlier, dictates that each term in the first set of parentheses must be multiplied by each term in the second set.
Step 1: Distribute 'd' across the second parentheses
We begin by multiplying the term 'd' from the first parentheses by each term in the second parentheses:
- d * 8d² = 8d³
- d * -5 = -5d
Step 2: Distribute '-7' across the second parentheses
Next, we multiply the term '-7' from the first parentheses by each term in the second parentheses:
- -7 * 8d² = -56d²
- -7 * -5 = 35
Step 3: Combine the resulting terms
Now, we gather all the terms we've obtained through the distributive process:
8d³ - 5d - 56d² + 35
Step 4: Arrange the terms in standard form
To present the expression in its conventional form, we arrange the terms in descending order of their exponents:
8d³ - 56d² - 5d + 35
This methodical approach ensures that no term is overlooked and that the final expression is in its simplest, most organized form. By meticulously following these steps, you can confidently expand and simplify polynomial expressions, laying a strong foundation for more advanced algebraic concepts.
Analyzing the Options
Now that we have meticulously expanded and simplified the expression (d - 7)(8d² - 5), arriving at the equivalent form 8d³ - 56d² - 5d + 35, it's crucial to compare our result with the provided options. This analytical step is not merely about identifying the correct answer; it's about reinforcing our understanding of the expansion process and honing our ability to recognize equivalent expressions.
Let's examine each option in light of our derived expression:
(A) 8d³ - 15d² - 5d + 35: This option deviates from our result in the coefficient of the d² term. It presents -15d² instead of -56d², indicating a potential error in the multiplication or combination of terms.
(B) 8d³ - 20d² + 56: This option is significantly different from our derived expression. It lacks the '-5d' term and has an incorrect constant term. It also has the wrong coefficient for the d² term. This suggests a fundamental misunderstanding of the expansion process.
(C) 8d³ - 56d² - 5d + 35: This option perfectly matches our derived expression. The coefficients and signs of all terms align with our step-by-step solution, confirming its equivalence to the original expression.
(D) 8d³ - 61d² + 56: This option also deviates from our result, particularly in the coefficient of the d² term. It presents -61d² instead of -56d². There are also significant differences with the constant. This discrepancy highlights the importance of careful multiplication and term combination.
Through this meticulous analysis, we can confidently affirm that option (C) is the only expression equivalent to (d - 7)(8d² - 5). This process not only provides the correct answer but also reinforces the value of verifying results and understanding the nuances of algebraic manipulations.
Correct Answer
After a thorough step-by-step expansion and simplification of the expression (d - 7)(8d² - 5), and a meticulous comparison with the provided options, we have arrived at a definitive conclusion. The equivalent expression is:
(C) 8d³ - 56d² - 5d + 35
This answer aligns perfectly with our derived expression, obtained through the correct application of the distributive property and careful combination of like terms. The other options presented variations in coefficients and terms, highlighting the importance of precise algebraic manipulation.
By selecting option (C), we demonstrate a firm grasp of polynomial expansion and simplification techniques. This understanding is not merely about arriving at the correct answer; it's about building a solid foundation for more advanced algebraic concepts and problem-solving strategies. The ability to confidently manipulate algebraic expressions is a crucial skill in mathematics and its applications, and this exercise has served to reinforce that skill.
Conclusion
In conclusion, the process of determining the equivalent expression for (d - 7)(8d² - 5) has been a valuable exercise in algebraic manipulation. Through the application of the distributive property and careful combination of like terms, we successfully expanded and simplified the expression to 8d³ - 56d² - 5d + 35.
This result directly corresponds to option (C), which stands as the correct answer. The other options presented variations that underscored the importance of precision in algebraic operations. This exercise not only reinforces the specific skill of polynomial expansion but also highlights the broader principles of algebraic problem-solving.
The ability to confidently manipulate algebraic expressions is a cornerstone of mathematical proficiency. It empowers us to solve equations, simplify complex relationships, and gain deeper insights into mathematical concepts. By mastering these techniques, we equip ourselves with the tools necessary to tackle a wide range of mathematical challenges and applications.
This exploration serves as a reminder that mathematics is not merely about memorizing formulas; it's about understanding the underlying principles and applying them strategically. With a solid foundation in algebraic manipulation, we can approach mathematical problems with confidence and clarity, paving the way for continued success in our mathematical endeavors.
