Problem 1 - Elementary Algebra

Question

Factor \(4 x-48\).

Step-by-Step Solution

Answer

Answer: The factored form of the expression is \(4(x-12)\).

Step 1: Identify the Greatest Common Factor (GCF)

In the expression \(4x - 48\), the coefficients are \(4\) and \(-48\). The greatest common factor of these coefficients is \(4\). We will now factor out the GCF from the expression.

Step 2: Factor out the GCF

To factor out the GCF, we divide each coefficient by the GCF and rewrite the expression. In this case, we will divide \(4x\) by \(4\) and \(-48\) by \(4\). This yields: \((4x / 4) - (48 / 4)\).

Step 3: Simplify the Expression

Now, we will simplify the result from Step 2: \((4x / 4) - (48 / 4)\) becomes \((x) - (12)\).

Step 4: Rewrite the factored expression

Lastly, we rewrite the factored expression with the GCF factored out. Our simplified expression from Step 3 is \((x) - (12)\), so we rewrite it with the GCF factored out: \(4(x-12)\). The factored expression is \(4(x-12)\).

Key Concepts

Understanding the Greatest Common Factor

The greatest common factor (GCF), also known as the greatest common divisor, is a fundamental concept in algebra that serves as the backbone for simplifying algebraic expressions. It is the highest number that divides exactly into two or more numbers without leaving a remainder.

For instance, when factoring the expression \(4x-48\), we first need to find the GCF of the numerical coefficients, which in this case are \(4\) and \(48\). To determine the GCF, we list the factors of each coefficient:

• Factors of \(4\): \({1, 2, 4}\)
• Factors of \(48\): \({1, 2, 3, 4, 6, 8, 12, 16, 24, 48}\)

The largest factor that appears in both lists is \(4\), making it the GCF of \(4\) and \(48\). Factoring out the GCF from the original expression simplifies it and often makes it more manageable to work with.

Decoding Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations such as addition, subtraction, multiplication, and division. In the example \(4x-48\), the expression includes the variable \(x\), the constant \(48\), and the operations of multiplication and subtraction.

When we look at such an expression, our goal is often to factor it, meaning to break it down into a product of simpler expressions. The motivation behind this process is not just to follow mathematical procedure but to unveil structural relationships within the algebraic expression and prepare it for further manipulation, such as solving equations.

The Art of Simplifying Expressions

Simplifying expressions is much like tidying up a cluttered room: we aim to clean up the algebraic expression so that it's easier to work with. The process involves several steps, including factoring out the GCF as described and then reducing any fractions or terms to their simplest form.

In our example, after the GCF of \(4\) is factored out, the expression inside the parentheses becomes much simpler: \(x-12\). This simplified expression enables easier further mathematical processes, such as solving related equations or analyzing functions. By mastering simplification techniques, students can navigate complex algebraic terrain with greater ease and confidence.