Problem 91 - Algebra A Combined Function

Question

Factor each expression completely. $$ a^{2}(b-4)-16(b-4) $$

Step-by-Step Solution

Answer

The expression factors completely to \((b-4)(a-4)(a+4)\).

Step 1: Identify Common Factor

First, observe that the expression \( a^2(b-4) - 16(b-4) \) possesses the common factor \((b-4)\) in both terms. Recognizing this allows us to factor it out from the expression.

Step 2: Factor Out the Common Factor

Extract \((b-4)\) from both terms of the expression: \[ a^2(b-4) - 16(b-4) = (b-4)(a^2 - 16) \].

Step 3: Recognize the Difference of Squares

Notice that the expression \( a^2 - 16 \) is a difference of squares. It can be factored using the formula \( x^2 - y^2 = (x - y)(x + y) \). In this case, \( x = a \) and \( y = 4 \) since \( 16 = 4^2 \).

Step 4: Factor the Difference of Squares

Divide \( a^2 - 16 \) using the difference of squares approach: \[ a^2 - 16 = (a - 4)(a + 4) \].

Step 5: Combine the Factors

Now rewrite the expression with the factored terms from the previous steps: \[ (b-4)(a-4)(a+4) \]. This is the expression fully factored.

Key Concepts

Common Factors

When dealing with algebraic expressions, identifying common factors is a fundamental technique in the factoring process. A common factor is simply a term that appears in each part of the expression.
In the exercise provided, the expression is \( a^2(b-4) - 16(b-4) \). Here, the term \((b-4)\) is a common factor as it appears in both parts of the expression.

The strategy for factoring out a common factor is to:

• Identify the term common across all parts of the expression. In this case, it's \((b-4)\).
• Use this common factor as a single 'group' or term by factoring it out of the entire expression. This simplifies the original expression significantly, reducing \( a^2(b-4) - 16(b-4) \) into \((b-4)(a^2 - 16)\).

Recognizing and extracting common factors simplifies expressions and paves the way for further factoring techniques.

Difference of Squares

The difference of squares is a special type of algebraic expression that can be factored very easily using a simple formula.
This type of expression always takes the form \( x^2 - y^2 \), and can be factored into \( (x-y)(x+y) \).

In the given problem, we're tasked with factoring \( a^2 - 16 \). Observing this, we know:

• \( a^2 \) is \( x^2 \) with \( x = a \).
• 16 is a square number, \( y^2 \) with \( y = 4 \).

Realizing this, \( a^2 - 16 \) can be rewritten and factored as \( (a - 4)(a + 4) \).
Recognizing structures like the difference of squares helps break down expressions into manageable, simpler pieces.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operators (like addition and multiplication).
These expressions are the building blocks of algebra, allowing for complex mathematical ideas to be represented and manipulated.

In factoring, each expression can often be broken down or transformed to make computations simpler. What's crucial is understanding:

• The role of each term in the expression, which involves both the numbers and the variables multiplied together.
• Navigating through expressions using mathematical rules and techniques, like factoring common factors and recognizing patterns such as the difference of squares.

Mastering the art of managing algebraic expressions by factoring allows for simplification and solution of problems that involve variables and their relationships, preparing you for more advanced algebraic concepts.