91Ó°ÊÓ

The distributive property is a cornerstone of algebra that allows us to multiply a single term by each term within a parenthesis. Simplifying an expression like \( (x+2)(3x-1) \) involves applying this property. To do this, you multiply the term outside the parentheses, 'x+2', by each term inside the other parenthesis, '3x' and '-1', one by one.

Similarly, you distribute '3x' across '(x-2)'. The distributive property is what helps us turn \( (x+2)(3x-1) \) into \( 3x^2 - x \) and \( 3x(x-2) \) into \( 3x^2 - 6x \) in the first step of the solution. Without it, we couldn't simplify complex algebraic expressions effectively.

After using the distributive property, the next step is often to combine like terms. Like terms are terms that have the exact same variables raised to the same powers. In other words, their only difference is the coefficient—that number out in front. In our example, \(3x^2 - x \) and \(3x^2 - 6x\) are composed of like and unlike terms.

To combine like terms, we match them up and sum their coefficients. For instance, \(3x^2 \) and \(3x^2\) are like terms because they both contain \(x^2\), so you add their coefficients to get \(6x^2\). On the other hand, \( -x\) and \( -6x\) are also like terms, and when you add them together, you get \( -7x\). This process simplifies \(3x^2 - x + 3x^2 - 6x\) into \(6x^2 - 7x\).

The process of simplifying algebraic expressions involves applying a series of steps, including the distributive property and combining like terms, to convert complex expressions into a more manageable form. The goal is to make the expression as simple as possible by eliminating parentheses and combining terms.

The example provided showcases this simplification process. After distributing and combining like terms, we are left with the simplified form \(6x^2 - 7x\). Simplification can make it easier to work with equations, to solve for variables, and to analyze mathematical relationships. Moreover, understanding this process is essential for success in more advanced algebra and calculus topics.