Polynomial Multiplication Explained Step-by-Step

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In the realm of mathematics, polynomial multiplication stands as a fundamental operation, forming the bedrock for more advanced algebraic manipulations. This article delves into the intricacies of multiplying polynomials, offering a step-by-step approach to solving the given expression: (a2)(2a3)(a^2 - 8a + 9). We will meticulously explore the process, ensuring a clear understanding of each stage, and ultimately arrive at the correct product.

Understanding Polynomial Multiplication

Before we embark on the solution, let's first grasp the essence of polynomial multiplication. A polynomial is an expression comprising variables and coefficients, combined using addition, subtraction, and multiplication, where the exponents of the variables are non-negative integers. Multiplying polynomials involves distributing each term of one polynomial across all terms of the other polynomial, and then combining like terms to obtain the simplified product.

Step-by-Step Solution

Now, let's tackle the given expression: (a2)(2a3)(a^2 - 8a + 9). To find the product, we'll follow these steps:

1. Multiply the first two terms:

We begin by multiplying the first two terms, (a^2) and (2a^3). This involves multiplying the coefficients and adding the exponents of the variable 'a':

(a2)(2a3) = 2a^(2+3) = 2a^5

This step utilizes the fundamental rule of exponents: x^m * x^n = x^(m+n). By applying this rule, we simplify the product of the first two terms to 2a^5. This resulting term will then be multiplied by the remaining polynomial.

2. Distribute the result across the remaining polynomial:

Next, we take the result from the previous step, 2a^5, and distribute it across the remaining polynomial, (a^2 - 8a + 9). This means multiplying 2a^5 by each term within the parentheses:

2a5(a2 - 8a + 9) = (2a^5 * a^2) - (2a^5 * 8a) + (2a^5 * 9)

This step is crucial as it expands the expression, setting the stage for combining like terms. The distributive property, a cornerstone of algebra, ensures that each term within the polynomial is multiplied by the term outside the parentheses.

3. Simplify each term:

Now, we simplify each term obtained in the previous step, again using the rule of exponents:

  • (2a^5 * a^2) = 2a^(5+2) = 2a^7
  • (2a^5 * 8a) = 16a^(5+1) = 16a^6
  • (2a^5 * 9) = 18a^5

Each term is simplified by multiplying the coefficients and adding the exponents of the variable 'a'. This process transforms the expanded expression into a collection of individual terms, each with a specific coefficient and exponent.

4. Combine the simplified terms:

Finally, we combine the simplified terms to obtain the final product:

2a^7 - 16a^6 + 18a^5

This step involves writing the simplified terms together, maintaining their respective signs. The result is the final product of the original expression, representing the polynomial obtained after multiplication and simplification.

Identifying the Correct Answer

Comparing our result, 2a^7 - 16a^6 + 18a^5, with the given options:

A. 2a^7 - 16a^6 + 18a^5 B. 2a^7 - 16a^6 - 18a^5 C. 2a^8 - 16a^7 + 18a^6 D. 2a^12 - 16a^7 + 18a^6

We can clearly see that option A matches our calculated product. Therefore, the correct answer is:

A. 2a^7 - 16a^6 + 18a^5

Key Concepts and Considerations

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. Understanding the degree can help predict the degree of the resulting polynomial after multiplication. In this case, the original expression involves polynomials of degrees 2, 3, and 2. The product will have a degree equal to the sum of the degrees of the individual polynomials, which is 2 + 3 + 2 = 7. This aligns with the degree of our final product, 2a^7 - 16a^6 + 18a^5.

Common Mistakes to Avoid

  • Forgetting to distribute: A common error is failing to distribute all terms properly. Ensure that each term in one polynomial is multiplied by every term in the other polynomial.
  • Incorrectly applying exponent rules: Mistakes in adding exponents can lead to incorrect results. Remember the rule x^m * x^n = x^(m+n).
  • Combining unlike terms: Only like terms (terms with the same variable and exponent) can be combined. For example, 2a^7 and 16a^6 cannot be combined.

Importance of Polynomial Multiplication

Polynomial multiplication is not just an abstract mathematical concept; it has practical applications in various fields, including:

  • Engineering: Used in circuit analysis, signal processing, and control systems.
  • Computer science: Employed in algorithm design, cryptography, and data compression.
  • Physics: Utilized in modeling physical phenomena, such as projectile motion and wave propagation.
  • Economics: Applied in modeling economic growth and financial markets.

The ability to multiply polynomials accurately is crucial for solving problems in these domains.

Practice Problems

To solidify your understanding, try solving these practice problems:

  1. (x + 2)(x - 3)(x + 1)
  2. (2y^2 - y + 1)(y^2 + 3y - 2)
  3. (a^3 + 2a^2 - a)(a^2 - 4a + 3)

Work through these problems step-by-step, paying close attention to distribution, exponent rules, and combining like terms. The solutions can be found online or in algebra textbooks.

Conclusion

Mastering polynomial multiplication is essential for success in algebra and beyond. By understanding the underlying principles, following a systematic approach, and practicing regularly, you can confidently tackle any polynomial multiplication problem. In this article, we have thoroughly explored the process of multiplying polynomials, focusing on the expression (a2)(2a3)(a^2 - 8a + 9). We have broken down the steps, explained the key concepts, and highlighted common mistakes to avoid. With this knowledge, you are well-equipped to navigate the world of polynomial multiplication and its diverse applications.

This comprehensive guide serves as a valuable resource for students, educators, and anyone seeking to enhance their understanding of polynomial multiplication. By grasping the fundamentals and practicing diligently, you can unlock the power of algebraic manipulation and apply it to a wide range of mathematical and real-world challenges. Remember, the key to success lies in consistent practice and a deep understanding of the underlying principles. So, continue exploring, experimenting, and mastering the art of polynomial multiplication!