Expanding Binomials A Step-by-Step Solution For (6b+7)(8b+8)

In the realm of mathematics, expanding algebraic expressions is a fundamental skill. Today, we embark on a journey to dissect and simplify the product of two binomials: (6b+7)(8b+8). This seemingly simple expression holds a wealth of mathematical concepts, from the distributive property to the art of combining like terms. Our goal is not merely to arrive at the final answer, but to unravel the underlying process, ensuring a comprehensive understanding that empowers you to tackle similar challenges with confidence.

Understanding the Distributive Property: The Key to Expansion

The cornerstone of expanding binomial products lies in the distributive property. This property dictates how we multiply a sum or difference by another term. In essence, it states that each term within the first set of parentheses must be multiplied by each term within the second set. To visualize this, let's break down our expression, (6b+7)(8b+8). We can view this as multiplying the binomial (6b+7) by each term in the binomial (8b+8), and vice versa. This can be written as:

6b * (8b + 8) + 7 * (8b + 8)

Now, we apply the distributive property again, this time within each of the resulting terms. This means multiplying 6b by both 8b and 8, and then multiplying 7 by both 8b and 8. This meticulous step-by-step approach ensures that we account for every possible product, leaving no term behind. The result is a series of individual multiplications that we can then simplify:

(6b * 8b) + (6b * 8) + (7 * 8b) + (7 * 8)

This expansion is the heart of the process. It transforms a compact expression into a sum of individual terms, each representing a specific product. From here, we can leverage our understanding of algebraic manipulation to simplify and combine these terms, ultimately arriving at the fully expanded form of the expression.

The Art of Multiplication: Unveiling the Individual Products

Having laid the groundwork with the distributive property, we now turn our attention to the individual multiplications that constitute the expanded expression. Each term represents the product of two components, and mastering these multiplications is crucial for achieving accuracy and efficiency. Let's delve into each term systematically:

• (6b * 8b): This term involves the product of two terms containing the variable 'b'. Recall that when multiplying variables with exponents, we multiply the coefficients and add the exponents. In this case, 6 multiplied by 8 yields 48, and 'b' multiplied by 'b' results in 'b^2' (b squared). Therefore, the product is 48b^2. This term represents a quadratic component, signifying the presence of a squared variable.

• (6b * 8): Here, we multiply a term with a variable by a constant. The process is straightforward: multiply the coefficient of the variable term (6) by the constant (8). This gives us 48, and we simply append the variable 'b' to the result. Thus, the product is 48b. This term represents a linear component, as the variable 'b' is raised to the power of 1.

• (7 * 8b): Similar to the previous term, this involves multiplying a constant by a term with a variable. We multiply 7 by 8, obtaining 56, and then append the variable 'b'. The resulting product is 56b. This is another linear term, contributing to the overall linear component of the expanded expression.

• (7 * 8): This term represents the simplest form of multiplication – the product of two constants. Multiplying 7 by 8 gives us 56. This constant term plays a crucial role in the final simplified expression, representing the numerical offset.

Each of these individual products contributes to the overall expanded expression. The next step involves carefully combining these terms, leveraging our understanding of like terms and their properties.

Combining Like Terms: The Path to Simplification

With the individual products calculated, we arrive at a crucial stage in the expansion process: combining like terms. Like terms are those that share the same variable raised to the same power. In our expanded expression, 48b^2 + 48b + 56b + 56, we can readily identify two terms that fit this criterion: 48b and 56b. Both terms contain the variable 'b' raised to the power of 1, making them eligible for combination.

The act of combining like terms involves adding or subtracting their coefficients while keeping the variable part unchanged. In this case, we add the coefficients 48 and 56, resulting in 104. The variable part, 'b', remains the same. Therefore, the combined term is 104b. This simplification consolidates the linear components of the expression, making it more concise and easier to interpret.

The remaining terms, 48b^2 and 56, do not have any like terms within the expression. The term 48b^2 is a quadratic term, and there are no other quadratic terms to combine it with. Similarly, the term 56 is a constant term, and there are no other constant terms present. These terms remain as they are, contributing to the final form of the simplified expression.

The process of combining like terms is a fundamental technique in algebraic simplification. It allows us to reduce the complexity of expressions by grouping together terms that share common characteristics. This not only makes the expression more manageable but also reveals its underlying structure and relationships more clearly.

The Final Product: Unveiling the Simplified Expression

Having meticulously expanded the binomial product and skillfully combined like terms, we now stand at the culmination of our endeavor: the final, simplified expression. By bringing together the results of our previous steps, we arrive at the following:

48b^2 + 104b + 56

This expression represents the fully expanded and simplified form of the original binomial product, (6b+7)(8b+8). It is a quadratic expression, characterized by the presence of a squared term (48b^2), a linear term (104b), and a constant term (56). Each term plays a distinct role in defining the behavior and characteristics of the expression.

The quadratic term, 48b^2, dictates the overall shape of the expression's graph, which is a parabola. The coefficient 48 determines the parabola's concavity and steepness. The linear term, 104b, influences the parabola's position and slope. The constant term, 56, represents the y-intercept of the graph, the point where the parabola intersects the vertical axis.

This final expression is not merely an answer; it is a testament to the power of algebraic manipulation. It encapsulates the intricate relationships between the original binomials and provides a concise representation of their product. Understanding the components of this expression – the quadratic, linear, and constant terms – unlocks deeper insights into its mathematical properties and behavior.

Applications and Extensions: Beyond the Basics

The process of expanding binomial products extends far beyond the confines of textbook exercises. It is a fundamental skill that finds application in various mathematical domains, including calculus, algebra, and geometry. For instance, in calculus, expanding expressions is often a necessary step in differentiation and integration. In algebra, it is crucial for solving equations and simplifying complex expressions. In geometry, it can be used to calculate areas and volumes.

Moreover, the principles we have explored today can be generalized to more complex expressions involving multiple binomials or polynomials. The distributive property remains the cornerstone of the expansion process, and the art of combining like terms is equally crucial. As you encounter more challenging expressions, the systematic approach we have outlined will serve as a reliable guide.

In conclusion, the expansion of the binomial product (6b+7)(8b+8) is a journey into the heart of algebraic manipulation. By mastering the distributive property, diligently performing multiplications, and skillfully combining like terms, we have unveiled the final, simplified expression: 48b^2 + 104b + 56. This expression is not merely an answer; it is a gateway to a deeper understanding of mathematical concepts and their applications. So, embrace the power of expansion, and let it empower you to conquer mathematical challenges with confidence and precision.