Step-by-Step Solution Multiplying Polynomials -4b²(5b² - 3b - 2)

In the realm of mathematics, polynomial multiplication is a fundamental operation. It is essential for solving algebraic equations and simplifying expressions. This article will delve into the process of multiplying a monomial by a polynomial, using the example of -4b²(5b² - 3b - 2). We will break down each step, ensuring a clear and comprehensive understanding of the procedure. Mastering polynomial multiplication is a crucial stepping stone for tackling more advanced mathematical concepts, making it a valuable skill for students and professionals alike. So, let's embark on this mathematical journey and unravel the intricacies of multiplying polynomials.

Understanding the Distributive Property

The distributive property is the cornerstone of polynomial multiplication. This fundamental principle states that multiplying a single term by a group of terms enclosed in parentheses involves multiplying the single term by each term within the parentheses individually. It's like distributing a package to each person in a room. In mathematical terms, for any numbers a, b, and c, the distributive property can be expressed as:

• a(b + c) = ab + ac

This seemingly simple concept is the key to unlocking the process of polynomial multiplication. It allows us to break down complex expressions into smaller, more manageable components. By applying the distributive property, we can systematically multiply each term of the polynomial by the monomial, paving the way for simplification and solution. This property not only simplifies calculations but also provides a clear and logical framework for approaching polynomial multiplication problems. Understanding and mastering the distributive property is therefore paramount to success in algebra and beyond. Let's see how this property works in our example: -4b²(5b² - 3b - 2).

Step-by-Step Multiplication of -4b²(5b² - 3b - 2)

To solve the expression -4b²(5b² - 3b - 2), we will employ the distributive property. This involves multiplying the monomial -4b² by each term inside the parentheses: 5b², -3b, and -2. Let's break down each multiplication step by step:

1. Multiplying -4b² by 5b²

The first step involves multiplying -4b² by 5b². To do this, we multiply the coefficients (-4 and 5) and add the exponents of the variable b (2 and 2). The process is as follows:

• (-4) * (5) = -20
• b² * b² = b^(2+2) = b⁴

Therefore, the result of multiplying -4b² by 5b² is -20b⁴. This step showcases the application of basic multiplication rules combined with the exponent rule for multiplying powers with the same base. By carefully combining the coefficients and variables, we arrive at the first term of our simplified expression. Understanding this step is crucial as it sets the foundation for the subsequent multiplications and the final simplification of the polynomial expression.

2. Multiplying -4b² by -3b

Next, we multiply -4b² by -3b. Again, we multiply the coefficients and add the exponents of the variable b. Remember that the variable b without an explicit exponent has an exponent of 1. The multiplication process is:

• (-4) * (-3) = 12
• b² * b = b^(2+1) = b³

Thus, the result of multiplying -4b² by -3b is 12b³. Note the importance of paying attention to the signs; the product of two negative numbers is positive. This step further illustrates the combination of numerical and variable multiplication, highlighting the rules of exponents and sign conventions. A thorough grasp of these rules is essential for accurate polynomial multiplication. As we move to the next term, the consistency in applying these rules will ensure the correct simplification of the entire expression.

3. Multiplying -4b² by -2

Finally, we multiply -4b² by -2. This step involves multiplying the coefficient of the monomial by the constant term. The process is straightforward:

• (-4) * (-2) = 8
• b² remains as b² since there is no variable term to multiply it with.

Therefore, the result of multiplying -4b² by -2 is 8b². Again, the product of two negative numbers yields a positive result. This final multiplication completes the distribution process, giving us all the individual terms that will form our simplified polynomial. By carefully executing each step, we have successfully multiplied the monomial by each term of the polynomial, setting the stage for the final step of combining these terms into a single, simplified expression.

Combining the Results

After multiplying -4b² by each term inside the parentheses, we have the following results:

• -4b² * 5b² = -20b⁴
• -4b² * -3b = 12b³
• -4b² * -2 = 8b²

Now, we combine these results to form the simplified expression. This involves writing the terms together, paying attention to their signs. The combined expression is:

-20b⁴ + 12b³ + 8b²

This is the final simplified form of the original expression -4b²(5b² - 3b - 2). We cannot simplify this expression further because there are no like terms to combine. Like terms have the same variable raised to the same power. In our result, we have terms with b raised to the powers of 4, 3, and 2, which are distinct and cannot be combined. This final step underscores the importance of accurately tracking the results of each multiplication and combining them correctly to arrive at the solution. The simplified expression represents the culmination of the distributive property and the rules of exponents, showcasing the power of these mathematical tools in simplifying complex expressions.

Conclusion

In conclusion, we have successfully performed the multiplication -4b²(5b² - 3b - 2), which resulted in the simplified expression -20b⁴ + 12b³ + 8b². This process involved a clear application of the distributive property, multiplying the monomial by each term within the parentheses, and then combining the resulting terms. Each step, from multiplying the coefficients to adding the exponents, was carefully executed to ensure accuracy. The importance of paying attention to signs and applying the rules of exponents correctly was highlighted throughout the process. This example demonstrates the fundamental principles of polynomial multiplication, a crucial skill in algebra and higher-level mathematics. By mastering these techniques, students and professionals can confidently tackle more complex mathematical problems and applications. The ability to simplify and manipulate algebraic expressions is a cornerstone of mathematical proficiency, and polynomial multiplication is a key component of this skill set.