Problem 10 - Elementary Algebra

Question

List, if any appear, the common factors in the following expressions. $$ 9 a b(a-8)-15 a(a-8)^{2} $$

Step-by-Step Solution

Answer

Answer: The common factors in the given expression are 3, a, and \((a-8)\).

Step 1: Identify a common factor in the expression

We have the given expression: $$ 9ab(a-8)-15 a(a-8)^{2} $$ Notice that both terms have an a in common, as well as the term \((a-8)\). We can factor these common terms out of both terms in the expression.

Step 2: Factor out the common factors

We can use the distributive property to factor out the common factors, a and \((a-8)\), from both terms: $$ a(a-8)(9b-15(a-8)) $$

Step 3: Identify and simplify any further common factors

Now let's analyze the remaining part: $$(9b-15(a-8))$$ Both terms have a factor of 3 in common; let's factor that out as well: $$ 3(3b-5(a-8)) $$

Step 4: Put everything together

Now, let's replace the factored expression back into the original factored expression: $$ a(a-8)(3)(3b-5(a-8)) = 3a(a-8)(3b-5(a-8)) $$ So the common factors in the given expression are 3, a and \((a-8)\).

Key Concepts

Common factors

Understanding common factors is key to simplifying algebraic expressions. A common factor is a term that appears in two or more parts of an expression. For example, in this exercise, observe the expression: \(9ab(a-8) - 15a(a-8)^2\). Both terms share certain elements:

• "\(a\)" - appears in both terms.
• "\((a-8)\)" - also present twice.

Identifying these shared components allows us to factor them out. This process reduces an expression to its simplest form by expressing it in terms where these factors are isolated and highlighted.
Recognizing common factors might seem trivial at first, but it plays a crucial role in many algebra-related topics, especially when solving equations or proving identities. Always train your eyes to spot these elements as they appear throughout the terms.

Distributive property

The distributive property is an essential tool that enables us to simplify expressions by multiplying a single factor across terms inside a parenthesis. In algebra, the distributive property can be written as \(a(b + c) = ab + ac\).
In our exercise, after identifying the common factors, we used the distributive property to pull out "\(a\)" and "\((a-8)\)", transforming the expression into \(a(a-8)(9b - 15(a-8))\).
Distributing allows for neat and organized handling of algebraic formulas. By considering all terms affected by a common factor together, efficiency and clarity emerge, converting potentially complex problems into straightforward calculations.

Algebraic expressions

Algebraic expressions consist of variables and constants combined using arithmetic operations. Unlike equations, these expressions do not have an equality sign dictating left and right sides. The expression in our original exercise, \(9ab(a-8) - 15a(a-8)^2\), is a perfect example.
These expressions can often initially appear unmanageable. However, through a deeper understanding of techniques like factoring and using the distributive property, we can distill them down. Simplifying such expressions is instrumental in solving more complex problems later on.
When working with algebraic expressions, focusing on structure and the relationships between components can reveal steps for simplification or other manipulations, promoting overall understanding. Being able to maneuver in this symbolic language of algebra is fundamental in mathematics.