Problem 5 - Elementary and Intermediate Algebra

Question

Fill in the blanks. To factor \(a b+6 a+2 b+12\) by ____, we begin by factoring out \(a\) from the first two terms and 2 from the last two terms.

Step-by-Step Solution

Answer

Factor by grouping to get \((a + 2)(b + 6)\).

Step 1: Identify Part I Terms

The first part of the expression is \(ab + 6a\). Identify \(a\) as a common factor in these terms.

Step 2: Factor Out Part I

Factor \(a\) out of the terms \(ab + 6a\) to get \(a(b + 6)\).

Step 3: Identify Part II Terms

The second part of the expression is \(2b + 12\). Identify \(2\) as a common factor in these terms.

Step 4: Factor Out Part II

Factor \(2\) out of the terms \(2b + 12\) to get \(2(b + 6)\).

Step 5: Recognize Common Binomial Factor

Both factored parts contain the common binomial factor \((b + 6)\).

Step 6: Perform Final Factoring

Combine the common factor \((b + 6)\) from the two parts to write the expression as \((a + 2)(b + 6)\). Thus, the expression is factored completely.

Key Concepts

Common Factors

When working with algebraic expressions, one of the key skills is identifying common factors. This means finding a number or variable that is present in multiple terms of the expression.
In our exercise, the expression is divided into two parts: \(ab + 6a\) and \(2b + 12\).

• Observe that both terms in \(ab + 6a\) share \(a\) as a common factor.
• In \(2b + 12\), the number \(2\) is the common factor in these terms.

To factor an expression, identify the common factor and "factor it out," which means rewriting the expression to make this common factor explicit.
This process simplifies the expression and is a crucial step in solving and simplifying algebraic problems.

Binomial Expressions

In algebra, a binomial expression is a polynomial with precisely two terms. In this exercise, a common binomial expression is found in the parentheses after factoring out the common numbers in both parts.
After factoring, the expression becomes \(a(b + 6) + 2(b + 6)\). Notice the binomial \((b + 6)\) is present in both terms.

• Binomials can be easily identified by their format: two terms connected by a plus or minus sign.
• Understanding how to manipulate binomials is crucial for successful factoring and algebraic simplification.

Recognizing recurring binomial expressions within an algebraic expression is essential, as it often indicates that further factoring or simplification is possible.
This leads to a more compact and manageable form of the original expression.

Distributive Property

The distributive property is a fundamental tool in algebra, allowing you to multiply a single term across terms inside a parenthesis. It is expressed as \(a(b + c) = ab + ac\). In factoring, this property works in reverse.
Once you have the factored terms such as \(a(b + 6)\) and \(2(b + 6)\), you can use the distributive property to combine them using their common binomial factor \((b + 6)\).

• The common binomial \((b + 6)\) can be factored out of both, resulting in \((a + 2)(b + 6)\).
• This step showcases the distributive process being reversed, where factors are brought together instead of spread apart.

Using the distributive property is an efficient way to simplify expressions and solve equations, making it fundamental to mastering algebraic manipulations.