91Ó°ÊÓ

A binomial is an algebraic expression that contains two terms. Examples include \(x + 2\) and \(x - 5\). These terms can be constants or variables, and they are often connected by a plus or minus sign. In our exercise, we are working with the binomials \(x + 2\), \(x + 4\), and \(x - 5\). Each of these pairs of terms comes together to help form a polynomial expression that we will be expanding and simplifying. Understanding binomials is key because it serves as a foundation for more complex polynomial manipulations.

Expanding an expression involves multiplying the terms inside the parentheses to form a larger polynomial. For instance, if we take \( (x+2)(x+4) \), we use the distributive property (often called FOIL for First, Outer, Inner, Last when dealing with binomials) to multiply each term from the first binomial by each term in the second binomial. This yields: \[ (x+2)(x+4) = x \cdot x + x \cdot 4 + 2 \cdot x + 2 \cdot 4 = x^2 + 4x + 2x + 8 = x^2 + 6x + 8\] Once we have this result, we can expand it further by multiplying it with the third binomial \(x - 5\). We follow similar steps: \[ (x^2 + 6x + 8)(x-5) = x^2(x-5) + 6x(x-5) + 8(x-5)= x^3 - 5x^2 + 6x^2 - 30x + 8x - 40\] The goal of expanding is to turn products of simpler expressions into a single, often larger, polynomial.

Once the expression is expanded, the next step is to simplify it by combining like terms. Like terms are terms that have the same variable raised to the same power. In the polynomial from our example, \( x^3 - 5x^2 + 6x^2 - 30x + 8x - 40 \), we combine the like terms: \[ x^3 + (6x^2 - 5x^2) + (-30x + 8x) - 40 = x^3 + x^2 - 22x - 40\] Here, \x^3\ remains the same as there are no other cubes of x to combine with. For the \x^2\ terms, \(6x^2 - 5x^2\) simplifies to \x^2\. For the x terms, \(-30x + 8x \) simplifies to \ -22x \. Constants are left as is unless there are other constants to combine with. The final expression, therefore, is \ x^3 + x^2 - 22x - 40 \.