91Ó°ÊÓ

The difference of squares is a powerful algebraic concept that simplifies certain types of expressions. When you encounter an expression like

• \((a+b)(a-b)\),

you are looking at a classic case of a difference of squares formula at work.

**Understanding the Formula**
The formula for the difference of squares states:

• \((a-b)(a+b) = a^2 - b^2\).

In this scenario, the expression within the parenthesis can be multiplied directly to result in the square of the first term minus the square of the second term. It provides an immediate shortcut to simplifying expressions without multiplying each term step by step.

Instead of distributing each term individually from one binomial to the other, recognizing a difference of squares allows you to jump immediately to a cleaner, simplified expression. In our original problem, applying the difference of squares simplifies the expression

• from \((a+b)(a-b)(a-3b)\) to \((a^2 - b^2)(a-3b)\),

which is one step closer to being solved.

The distributive property is a fundamental algebraic principle that lets you simplify and solve equations by handling multiplication across a sum or difference. When distributing, each term inside a parenthesis is multiplied by a term outside the parenthesis.

**Applying the Distributive Property**
To apply this concept, imagine you have an expression like:

• \((a^2 - b^2)(a-3b)\),

and you need to simplify it.

Distribute the terms sequentially:

• First, distribute \(a^2\) to both terms within the \((a-3b)\) binomial, giving you \(a^2 \times a = a^3\) and \(a^2 \times -3b = -3a^2b\).
• Next, distribute \(-b^2\) similarly, leading to \(-b^2 \times a = -ab^2\) and \(-b^2 \times -3b = 3b^3\).

Each step cuts through complexity by distributing multiplication and helps in reorganizing the terms for easier modifications later. Efficiency in these operations allows the expression to be rewritten in standard polynomial form quickly.

Combining like terms is an essential strategy in algebra used to simplify expressions and solve equations.

**Identifying Like Terms**
Like terms are terms that have the same variable raised to the same power. In the expression

• \(a^3 - 3a^2b - ab^2 + 3b^3\),

one might initially search for common factors or similar terms.

**Combining Process**
In this example, you notice that each term is distinct in terms of its variable components and powers:

• \(a^3\) is not similar to any other terms;
• \(-3a^2b\), \(-ab^2\), and \(3b^3\) are also unique.

While it's important to identify and combine terms that can be merged, this particular expression does not require further simplification beyond listing each term as part of the final simplified polynomial.

Combining like terms often results in fewer terms, making complex expressions more manageable and clear to understand or further process in equations.