Problem 49 - Elementary and Intermediate Algebra

Question

Factor out the GCF. $$ 24 a-16 a^{2} $$

Step-by-Step Solution

Answer

The expression factors to \(8a(3 - 2a)\).

Step 1: Identify Common Factors

Examine the terms in the expression to find any common factors. The expression is \(24a - 16a^2\). Notice both terms have a factor of \(a\) and a numerical GCF (Greatest Common Factor) of 8.

Step 2: Factor Out the Numerical GCF

From the terms \(24a\) and \(16a^2\), identify the numerical GCF, which is 8. Divide both terms by 8: \(\frac{24a}{8} = 3a\) and \(\frac{16a^2}{8} = 2a^2\), giving us \(8(3a - 2a^2)\).

Step 3: Factor Out the Variable GCF

Identify the variable GCF, which is \(a\). Factor \(a\) out from each term: \(3a - 2a^2 = a(3 - 2a)\).

Step 4: Combine the Factors

Combine the numerical and variable factors: The fully factored expression is \(8a(3 - 2a)\).

Key Concepts

Greatest Common Factor (GCF)

When tackling algebraic expressions, identifying the Greatest Common Factor (GCF) is a key skill that simplifies expressions. The GCF is the highest number or variable that divides each term in the expression without a remainder. For instance, in the expression \(24a - 16a^2\), we first need to consider both the numerical and variable parts.

The numerical part of each term is 24 and 16. The largest number that divides both 24 and 16 is 8. Therefore, 8 is the numerical GCF. For the variable part, each term contains the variable \(a\) and \(a^2\). The smallest power of the common variable, which is \(a^1\) (or just \(a\)), is the variable GCF. Together, these form the full GCF, which in this case is \(8a\).
Understanding GCF helps in breaking down complex expressions into simpler factors, making algebraic simplification more manageable.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols. They represent values and relationships and do not have an equals sign—unlike equations. In the expression \(24a - 16a^2\), there are two terms separated by a subtraction sign.

* Terms: These are parts of an expression separated by addition or subtraction signs. In our example, the terms are \(24a\) and \(16a^2\). * Numerical coefficients: These are the numbers multiplying the variables in a term. For \(24a\), 24 is the coefficient, and for \(16a^2\), it's 16. * Variables: These are letters that represent numbers in the expression. Here we have \(a\) and \(a^2\), where \(a\) is a variable, and \(a^2\) indicates \(a\) multiplied by itself.
When working with algebraic expressions, it's essential to correctly identify each component as this understanding forms the foundation for more advanced operations, like factoring.

Factoring Techniques

Factoring techniques are methods used to express an algebraic expression as a product of simpler factors. This process makes equations easier to solve and understand. To factor expressions, follow these steps:

**1. Identify the GCF:** As shown in the previous example, you must first find the GCF. For \(24a - 16a^2\), the GCF is \(8a\).

**2. Divide Each Term by the GCF:** Dividing terms by their GCF allows us to write the expression in simpler terms. So, dividing \(24a\) by \(8a\) gives \(3\), and dividing \(16a^2\) by \(8a\) gives \(2a\).

**3. Write the Simplified Expression in Factored Form:** Combining these gives us the expression \(8a(3 - 2a)\). This is the fully factored form of the original expression.

Knowing how to factor expressions not only helps solve equations efficiently but also gives greater insight into the relationships between terms. Making use of GCF and properly applying these techniques simplifies the computational process significantly.