91Ó°ÊÓ

The distributive property is a fundamental rule in algebra that helps with multiplying expressions. In essence, it means distributing or spreading out a multiplier across terms inside parentheses.
For the expression \(a+2\)(a-2)\(a^2+4\), we applied the distributive property as follows:

• First, \(a+2\) multiplies by \(a-2\)\: \((a+2)(a-2) = a(a-2) + 2(a-2)\).
• Then, the result \(a^2-4\) multiplies by \(a^2+4\)\: \(a^2(a^2 + 4) - 4(a^2 + 4)\).

Using this property ensures each term in one factor multiplies every term in the other factor, leading to accurate expansions.
This principle is crucial in algebra and lays the foundation for more complex operations in polynomial equations.

The FOIL method is a specific application of the distributive property used primarily for multiplying two binomials. FOIL is an acronym for:

• First: Multiply the first terms in each binomial.
• Outside: Multiply the outer terms in the product.
• Inside: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

For the binomials \(a+2\) and \(a-2\), applying FOIL we calculate:

• ext{First: } a imes a = a^2
• ext{Outside: } a imes (-2) = -2a
• ext{Inside: } 2 imes a = 2a
• ext{Last: } 2 imes (-2) = -4

Combine these to get \(a^2 - 2a + 2a - 4\). When simplified, it becomes the difference of squares: \(a^2 - 4\).
FOIL is particularly useful for quick calculations with binomials, simplifying the process of polynomial expansion.

Combining like terms is an essential step in simplifying algebraic expressions. Like terms are terms with the same variable raised to the same power.
In our step-by-step solution, we encountered terms such as \(4a^2\) and \(-4a^2\). Since they both contain \(a^2\), these are classified as like terms:

• ext{Combine } 4a^2 ext{ and } -4a^2 ext{ to simplify as follows: } 4a^2 - 4a^2 = 0.

Identifying and combining like terms simplifies expressions by merging all terms that are alike into a single term.
This reduces clutter and makes equations or expressions cleaner, making it easier to analyze or further solve.

A quadratic expression is one of the simplest forms of polynomial expressions where the highest degree is 2. It usually appears in forms like \(ax^2 + bx + c\).
In our exercise, we initially worked with binomials \(a+2\) and \(a-2\) that, when expanded, formed a product leading to a quadratic: \(a^2 - 4\).
This difference of squares is a special quadratic that simplifies polynomial expressions by creating elegant and simplified versions of expanded forms.
Quadratic expressions are vital in algebra because they frequently come up in equations modeling real-world scenarios, such as projectile motion. Solving and simplifying quadratics form the cornerstone of many algebraic problem-solving techniques.