Problem 38 - Algebra and Trigonometry

Question

\(35-82\) Factor the expression completely. $$ 5 a b-8 a b c $$

Step-by-Step Solution

Answer

The expression factors to \(ab(5 - 8c)\).

Step 1: Identify Common Factors

First, observe the expression: \(5ab - 8abc\). The terms \(5ab\) and \(8abc\) have common factors. Inspect closely, and notice that both terms have \(a\) and \(b\) as factors. Identify \(ab\) as the common factor.

Step 2: Factor Out the Common Factor

Once the common factor \(ab\) is identified, factor it out from the expression. This will leave: \[ ab(5 - 8c) \].By factoring out \(ab\), we rewrite the expression in a simplified form.

Step 3: Verify the Factorization

To ensure the factorization is correct, distribute \(ab\) back through the expression inside the parentheses: \(ab(5 - 8c) = ab imes 5 - ab imes 8c = 5ab - 8abc\).The original expression and the expanded form match, so the factorization is correct.

Key Concepts

Common Factors

In algebra, recognizing common factors in an expression is an essential step to simplify or factor the expression. A common factor is an element that can be evenly divided out from each term of the expression without leaving a remainder. Let's take the example expression:

• 5ab - 8abc

Both terms of this expression, namely, 5ab and 8abc, have certain factors that are shared. By closely inspecting these terms, we notice that they both contain the variables a and b. These common factors, a and b, can be grouped together as ab, which is the greatest common factor (GCF) for both terms. Finding and factoring out this common factor is important as it simplifies the expression and helps to reveal its simplest form.

Algebraic Expression

An algebraic expression is a mathematical phrase that includes numbers, variables, and operators such as addition or multiplication. Unlike equations, algebraic expressions do not contain an equality sign. Consider the expression in our exercise:

• 5ab - 8abc

This expression comprises two terms. Each term is a combination of a coefficient and variables. A coefficient is a numerical value that multiplies the variables. Here, in the term 5ab, 5 is the coefficient, while a and b are the variables. Similarly, in the term 8abc, 8 is the coefficient, and a, b, c are the variables. Factoring this expression means rewriting it in a way that shows the common factors explicitly, simplifying the manipulation of complex algebra problems.

Factorization Verification

After factoring an expression, it's crucial to check that the factorization is correct. This step is often called factorization verification. We do this by distributing the factored terms back into the expression to see if it results in the original terms. Taking our simplified expression:

• \[ ab(5 - 8c) \]

To verify, we distribute the common factor ab through the expression inside the parentheses using the distributive property:

• \[ ab \times 5 = 5ab \]
• \[ ab \times -8c = -8abc \]

When combined, these give us the original expression:

• 5ab - 8abc

The result matches what we started with, confirming the factorization is done correctly. This verification step ensures accuracy and builds confidence in handling algebraic manipulations.