Problem 78 - Elementary Algebra

Question

Factor completely. $$ 8 \times 3 y 3-27 $$

Step-by-Step Solution

Answer

The expression factors to \( 3(8y - 9) \).

Step 1: Identify Common Factors

First, we need to identify the greatest common factor (GCF) of the terms in the expression. For the numbers 24 ( 8 imes 3 y ) and 27, the GCF is 3.

Step 2: Factor Out the GCF

Next, factor the GCF from both terms of the expression. The expression becomes: \[ 3(8y - 9) \]

Step 3: Verify Factored Expression

We have factored out the greatest common factor, and no further factoring is possible for the expression 8y - 9 , which indicates that the factorization is complete.

Key Concepts

Understanding the Greatest Common Factor

In algebra, finding the Greatest Common Factor (GCF) is a key step in simplifying expressions, especially when factoring. The GCF is the largest number that divides all the terms in an expression without leaving a remainder. This helps to simplify expressions and make calculations easier.
For example, in the exercise that involves factoring the expression \(8 \times 3y - 27\), we first identify the numbers involved, 24 (from \(8 \times 3y\)) and 27. By breaking these numbers into their prime factors:

• 24 can be factored into \(2^3 \times 3\)
• 27 can be factored into \(3^3\)

The common factor is 3, making it the GCF. By factoring out 3, we simplify the expression, setting us up for further manipulation if needed.

Factoring Binomials Explained

Once the GCF has been identified and factored out, the remaining expression is often a binomial. A binomial is a polynomial with two terms, like \(8y - 9\) in our case.
Factoring a binomial can sometimes be more complex, involving methods like difference of squares or applying the quadratic formula. However, in this instance of \(8y - 9\), further factoring isn't possible because:

• 8y and 9 have no common factors other than 1.
• The terms don't fit special patterns like difference of squares.

Thus, factoring the GCF is all that's required, leaving us with \(3(8y - 9)\) as the completely factored form.

Breaking Down Algebraic Expressions

Algebraic expressions like \(8 \times 3y - 27\) consist of variables, numbers, and operators that together form a mathematical phrase. They lack an equality sign and hence aren't equations.
To simplify or factor algebraic expressions:

• Identify and separate like terms, if any.
• Look for common factors or patterns.
• Apply factoring techniques to break down the expression as much as possible.

Understanding these expressions is essential for mastering many algebraic operations and solving more complex problems. By approaching them step-by-step, as we've demonstrated, you can gain confidence and skill in dealing with them effectively.