91Ó°ÊÓ

The Distributive Law is one of the cornerstones of algebra. It allows you to multiply a single term by each term within a parenthesis. This is especially useful when you have an expression where terms are grouped or formed as products. In this exercise, you'll see how the distributive law comes into play.To apply it, if you have an expression like \[a(b + c)\]You distribute the 'a' across each term within the parentheses:\[= ab + ac\]In the example expression, \[(2x)(5x)+(3x)(2x)+5(3x)+x(3x)\]This becomes \[10x^2, 6x^2, 15x, \text{ and } 3x^2\]This process of distributing helps break down and reorganize an expression to allow further simplification. It's the initial step before combining terms together.

Once you've applied the distributive law, the next step in simplifying an expression is to combine like terms. Like terms in algebra are terms that have the same variable and exponent. To combine like terms, you simply add or subtract the coefficients of these terms. For instance, in the expression \[10x^2 + 6x^2 + 15x + 3x^2\]you'll notice that there are multiple terms that contain \(x^2\). Simply add up their coefficients:\[10x^2 + 6x^2 + 3x^2 = 19x^2\]While the term \(15x\) stands alone since there's no other \(x\) term to combine it with. So the process of combining like terms results in a cleaner and more succinct expression:

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They may seem complex, but mastering their simplification allows you to solve problems more efficiently. An algebraic expression can take many forms, such as linear, quadratic, or polynomial based on the highest power of the variable. The given exercise involves polynomial expressions which are commonly encountered in algebra. The process:

• Starts with applying operations like the distributive law.
• Followed by combining like terms to reduce the expression to its simplest form.

These strategies not only help in solving equations but also in understanding how variables interact and relate in different mathematical contexts. Simplifying algebraic expressions, as shown in the example exercise, is foundational to progressing in algebra and is applied extensively in calculus and beyond.