What Is The Middle Term In The Simplified Product Of (x+3)(x-4)? A. X B. -x C. 3x D. -4x

A. x B. -x C. 3x D. -4x

Multiplying Binomials: A Step-by-Step Guide

When we encounter expressions like extbf{(x+3)(x-4)}, we're dealing with the multiplication of two binomials. Binomials, in mathematical terms, are algebraic expressions that consist of two terms. To effectively multiply these binomials, we employ a method known as the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This systematic approach ensures that each term in the first binomial is multiplied by each term in the second binomial, leading to the correct expansion of the expression.

To begin, let's break down the FOIL method and apply it to our specific problem, extbf{(x+3)(x-4)}. F stands for First, meaning we multiply the first terms of each binomial together. In this case, it's x * x, which results in x^2. This term forms the foundation of our expanded expression, setting the stage for the subsequent steps. extbf{O** represents Outer, where we multiply the outer terms of the binomials. Here, we multiply x from the first binomial and -4 from the second binomial, yielding -4x. This term introduces a linear component to our expression, showcasing the interaction between the variable and a constant. Next, I signifies Inner, instructing us to multiply the inner terms of the binomials. This involves multiplying 3 from the first binomial and x from the second binomial, resulting in 3x. Similar to the Outer term, this also contributes a linear component, further shaping the expression. Finally, L stands for Last, where we multiply the last terms of each binomial. We multiply 3 from the first binomial and -4 from the second binomial, giving us -12. This constant term completes the expansion process, providing a fixed value that influences the overall expression.

By meticulously applying the FOIL method, we transform the product of two binomials into a more expanded form, revealing the individual terms that constitute the expression. Each step, from multiplying the first terms to the last, contributes a distinct component to the final result. The systematic nature of the FOIL method ensures that no term is overlooked, leading to an accurate and complete expansion of the binomial product.

Combining Like Terms: Simplifying the Expression

After expanding the product of binomials using the FOIL method, the next crucial step is to simplify the resulting expression. This simplification process primarily involves identifying and combining like terms. In algebraic expressions, like terms are those that have the same variable raised to the same power. Combining these terms streamlines the expression, making it easier to understand and work with.

In our expanded expression, x^2 - 4x + 3x - 12, we can readily identify the like terms. The terms -4x and 3x both contain the variable x raised to the power of 1, making them like terms. To combine these terms, we simply add their coefficients. In this case, we add -4 and 3, which results in -1. Therefore, the combination of -4x and 3x simplifies to -x. This step effectively reduces the number of terms in the expression, bringing us closer to the simplified form.

The remaining terms in the expression, x^2 and -12, do not have any like terms. The term x^2 is the only term with the variable x raised to the power of 2, and the term -12 is a constant term with no variable component. Since these terms cannot be combined with any other terms in the expression, they remain unchanged in the simplified form. The presence of unlike terms highlights the diversity of components within the expression, each contributing uniquely to its overall value.

By carefully identifying and combining like terms, we distill the expanded expression into its most concise form. This process not only simplifies the expression but also reveals its underlying structure, making it easier to analyze and manipulate in subsequent mathematical operations. The combination of like terms is a fundamental technique in algebra, essential for simplifying complex expressions and solving equations.

Identifying the Middle Term: The Solution

Having simplified the expression extbf{(x+3)(x-4)} to x^2 - x - 12, we can now readily identify the middle term. In a quadratic expression of the form ax^2 + bx + c, the middle term is the term that contains the variable x raised to the power of 1. This term, often referred to as the linear term, plays a crucial role in determining the behavior and properties of the quadratic expression.

In our simplified expression, x^2 - x - 12, the term that fits this description is -x. This term consists of the variable x with a coefficient of -1. The coefficient indicates the magnitude and direction of the term's contribution to the overall expression. In this case, the negative coefficient signifies that the term subtracts from the expression's value, influencing its shape and position on a graph.

The middle term, -x, is sandwiched between the quadratic term x^2 and the constant term -12. Its position within the expression is not merely coincidental; it reflects the term's role in connecting the squared variable and the constant. The middle term acts as a bridge, influencing the relationship between these two components and shaping the overall characteristics of the expression.

Therefore, when we examine the simplified product of extbf{(x+3)(x-4)}, the middle term stands out as -x. This term, with its variable and coefficient, holds significant weight in the expression's structure and behavior. By correctly identifying the middle term, we gain a deeper understanding of the expression and its properties, paving the way for further analysis and problem-solving.

Therefore, the correct answer is B. ext{-x}.

Conclusion

In conclusion, by meticulously applying the FOIL method and combining like terms, we successfully simplified the product of the binomials extbf{(x+3)(x-4)} to x^2 - x - 12. Through this process, we identified the middle term as extbf{-x}, which corresponds to option B. This exercise underscores the importance of understanding algebraic manipulation techniques and the structure of polynomial expressions. Mastering these concepts is crucial for success in mathematics and related fields, enabling us to solve complex problems and gain deeper insights into mathematical relationships. The ability to accurately expand and simplify expressions, identify key terms, and apply these skills to problem-solving is a testament to one's mathematical proficiency and lays the foundation for future learning and exploration in the world of mathematics.