Multiplying Polynomials A Step-by-Step Guide To Finding The Product Of (d-9) And (2d^2 + 11d - 4)

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Polynomial multiplication is a fundamental concept in algebra, and mastering it is crucial for solving various mathematical problems. In this article, we will delve into the process of multiplying two polynomials: (d - 9) and (2d^2 + 11d - 4). Our goal is to find the product of these expressions and identify the correct answer from the given options. This exercise not only enhances our understanding of polynomial operations but also sharpens our problem-solving skills in algebra. Let's embark on this mathematical journey to unravel the solution step by step.

Polynomial Multiplication Explained

To find the product of (d - 9) and (2d^2 + 11d - 4), we need to apply the distributive property of multiplication. This means each term in the first polynomial (d - 9) must be multiplied by each term in the second polynomial (2d^2 + 11d - 4). This process involves careful attention to detail, especially when dealing with signs and exponents. The distributive property, in essence, allows us to break down a complex multiplication problem into a series of simpler multiplications, which we can then combine to arrive at the final answer. Understanding this property is key to successfully navigating polynomial multiplication.

Let's break down the multiplication process step by step:

  1. Multiply 'd' by each term in the second polynomial:

    • d * 2d^2 = 2d^3
    • d * 11d = 11d^2
    • d * -4 = -4d

  2. Multiply '-9' by each term in the second polynomial:

    • -9 * 2d^2 = -18d^2
    • -9 * 11d = -99d
    • -9 * -4 = 36

  3. Combine the results:

    • Now, we add all the terms we obtained in the previous steps: 2d^3 + 11d^2 - 4d - 18d^2 - 99d + 36

  4. Simplify by combining like terms:

    • Like terms are those that have the same variable raised to the same power. In this expression, we can combine the d^2 terms and the d terms: 2d^3 + (11d^2 - 18d^2) + (-4d - 99d) + 36

  5. Perform the addition and subtraction:

    • 11d^2 - 18d^2 = -7d^2
    • -4d - 99d = -103d

  6. Write the final expression:

    • Putting it all together, we get: 2d^3 - 7d^2 - 103d + 36

This detailed breakdown illustrates how each term is carefully multiplied and combined, ensuring that we arrive at the correct product. The distributive property is not just a mathematical rule; it's a systematic approach to handling complex expressions, and mastering it is essential for success in algebra.

Step-by-Step Solution: Multiplying (d - 9) and (2d^2 + 11d - 4)

To effectively multiply the polynomials (d - 9) and (2d^2 + 11d - 4), we will employ the distributive property, ensuring each term in the first polynomial is multiplied by each term in the second. This meticulous process guarantees we capture all necessary products and sets the stage for simplification. By following this step-by-step method, we can confidently navigate polynomial multiplication, a cornerstone of algebraic operations. The resulting expression will then be simplified by combining like terms, ultimately leading us to the correct answer. Let's proceed with the multiplication:

  1. Multiply 'd' by each term in (2d^2 + 11d - 4):

    • d * 2d^2 = 2d^3: When multiplying variables with exponents, we add the exponents. Here, d (which is d^1) multiplied by 2d^2 gives us 2d^(1+2) = 2d^3. This is a fundamental rule in algebra and is crucial for correctly multiplying polynomials. The coefficient (2) remains as it is, as we are only multiplying the variable terms in this step.
    • d * 11d = 11d^2: Similarly, multiplying d (or d^1) by 11d (or 11d^1) results in 11d^(1+1) = 11d^2. The exponent of d increases by one as we multiply it by another d term. Understanding this principle is essential for handling polynomial multiplication efficiently.
    • d * -4 = -4d: Multiplying d by -4 results in -4d. This is a straightforward multiplication where we simply attach the variable d to the constant -4. The negative sign is important to maintain, as it affects the overall sign of the term in the final expression.

  2. Multiply '-9' by each term in (2d^2 + 11d - 4):

    • -9 * 2d^2 = -18d^2: Here, we multiply the constant -9 by the term 2d^2. The result is -18d^2. The coefficient is obtained by multiplying -9 and 2, and the variable part d^2 remains unchanged. Paying attention to the signs is crucial in this step to avoid errors in the final result.
    • -9 * 11d = -99d: Multiplying -9 by 11d gives us -99d. The coefficient is the product of -9 and 11, which is -99, and the variable d remains as it is. This step is another instance where the sign plays a significant role in the correctness of the calculation.
    • -9 * -4 = 36: When we multiply -9 by -4, we get a positive result because the product of two negative numbers is positive. The result is 36. This simple rule of signs is a fundamental aspect of arithmetic and algebra and is essential for accurate calculations.

  3. Combine all the results:

    • Now, we add all the terms we obtained from the previous steps: 2d^3 + 11d^2 - 4d - 18d^2 - 99d + 36. This step involves carefully listing out all the terms we calculated. Accuracy is key here to ensure that no terms are missed or incorrectly transcribed. This comprehensive listing sets the stage for the next step, where we will simplify the expression by combining like terms. Ensuring all terms are accounted for is a critical part of the process.

  4. Simplify by combining like terms:

    • Like terms are those that have the same variable raised to the same power. In our expression, 2d^3 + 11d^2 - 4d - 18d^2 - 99d + 36, we identify the d^2 terms (11d^2 and -18d^2) and the d terms (-4d and -99d) as like terms. The term 2d^3 and the constant 36 do not have any like terms and will remain as they are in the simplified expression. The process of identifying like terms is a crucial step in simplifying polynomials, as it allows us to reduce the expression to its most basic form. This skill is fundamental in algebra and is used extensively in various mathematical contexts.

  5. Perform the addition and subtraction:

    • Combine the d^2 terms: 11d^2 - 18d^2 = -7d^2. This operation involves subtracting the coefficients of the d^2 terms. The result is -7d^2. Accurate arithmetic is essential in this step to ensure the correct coefficient is obtained. This combination simplifies the quadratic terms in the polynomial, bringing us closer to the final simplified expression.
    • Combine the d terms: -4d - 99d = -103d. Similarly, we combine the coefficients of the d terms. The result is -103d. This step simplifies the linear terms in the polynomial. Paying attention to the signs is crucial here, as incorrect handling of signs can lead to errors in the final answer. This combination is a key step in reducing the complexity of the polynomial expression.

  6. Write the final expression:

    • Putting it all together, we get: 2d^3 - 7d^2 - 103d + 36. This is the simplified form of the polynomial expression resulting from the multiplication of (d - 9) and (2d^2 + 11d - 4). The terms are arranged in descending order of the powers of d, which is a standard practice in algebra. This final expression is the culmination of all the steps we have taken, from applying the distributive property to combining like terms. The accuracy of this final result depends on the precision and care taken in each preceding step.

By meticulously following these steps, we have successfully multiplied the two polynomials and arrived at the simplified expression. This process highlights the importance of attention to detail and the correct application of algebraic principles. Each step, from distributing terms to combining like terms, plays a crucial role in obtaining the accurate final answer.

Identifying the Correct Answer

Now that we have calculated the product of (d - 9) and (2d^2 + 11d - 4) to be 2d^3 - 7d^2 - 103d + 36, we can confidently identify the correct answer from the given options. This step is crucial to ensure that our calculated result matches one of the provided choices. By carefully comparing our answer with each option, we can validate our solution and select the correct one. This process not only confirms our mathematical accuracy but also reinforces our understanding of polynomial multiplication and simplification. Let's proceed to match our result with the given options.

Looking at the options provided:

A. 2d^3 - 7d^2 - 103d + 36 B. 2d^3 - 7d^2 - 95d + 36 C. 2d^3 + 7d^2 - 95d + 36 D. 2d^3 + 7d^2 - 103d + 36

By comparing our calculated product, 2d^3 - 7d^2 - 103d + 36, with the given options, we can see that it matches option A exactly. This confirms that our step-by-step solution has led us to the correct answer. The matching process involves checking each term, including its sign and coefficient, to ensure complete agreement. This validation step is essential in any mathematical problem-solving scenario, as it provides assurance that the final answer is accurate and reliable. In this case, the precise match with option A reinforces our confidence in the correctness of our solution.

Therefore, the correct answer is:

A. 2d^3 - 7d^2 - 103d + 36

This final step solidifies our understanding of the problem and our ability to solve it accurately. Identifying the correct answer is not just about finding a match; it's about validating the entire process and ensuring that each step was executed correctly. This comprehensive approach to problem-solving is a valuable skill in mathematics and other fields.

Conclusion

In conclusion, we have successfully determined the product of the polynomials (d - 9) and (2d^2 + 11d - 4) through a detailed, step-by-step process. We began by understanding the principles of polynomial multiplication, emphasizing the distributive property as the cornerstone of our approach. Each term in the first polynomial was meticulously multiplied by each term in the second polynomial, ensuring that all possible products were accounted for. This process highlighted the importance of accuracy in both arithmetic and algebraic manipulation. We then combined like terms to simplify the resulting expression, a crucial step in presenting the final answer in its most concise form. The simplified product was found to be 2d^3 - 7d^2 - 103d + 36, which matched option A from the given choices. This successful resolution underscores the significance of a systematic approach to problem-solving in algebra. By breaking down a complex problem into manageable steps, we were able to navigate the multiplication and simplification process with confidence. The skills and techniques demonstrated in this article are fundamental to algebraic proficiency and will serve as a valuable foundation for tackling more advanced mathematical challenges. This exercise not only reinforces our understanding of polynomial multiplication but also cultivates a methodical approach to problem-solving, a skill that is applicable across various disciplines.