Problem 2 - College Algebra with Modeling and Visualization

Question

Factor out the greatest common factor:. \(32-16 x\)

Step-by-Step Solution

Answer

The expression factors to \(16(2 - x)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, look at the numbers and terms in the expression \(32 - 16x\). The numbers are 32 and 16. To find the GCF, identify the largest number that can evenly divide both 32 and 16. The number 16 is the largest factor common to both.

Step 2: Factor Out the GCF

Now that we know the GCF is 16, we can factor it out from each term in the expression \(32 - 16x\). Divide each term by the GCF and write the expression as a product: \[32 - 16x = 16(2 - x)\]. This means we take 16 out of both terms, leaving 2 and -x inside the parentheses.

Step 3: Write the Factored Expression

The factored form of the expression \(32 - 16x\) is \(16(2 - x)\). Double-check your work by distributing the 16 back into the terms within the parentheses to ensure it equals the original expression.

Key Concepts

Greatest Common Factor (GCF)

The concept of the Greatest Common Factor (GCF) plays a crucial role in simplifying algebraic expressions. To comprehend it, imagine the GCF as the largest possible number that can exactly divide two or more numbers without leaving a remainder. For the expression \(32-16x\), we start by identifying the numbers involved, which are 32 and 16. By finding common factors of these numbers, we observe which is the greatest.

• The factors of 32 are: 1, 2, 4, 8, 16, 32
• The factors of 16 are: 1, 2, 4, 8, 16

By comparing these, we establish that the largest common factor is 16. This number will be useful for simplifying the expression via factoring. Recognizing the GCF efficiently allows the decomposition of the expression with ease, leading to simpler calculations and a clearer representation of the original algebraic statement.

Factored Form

Factoring involves breaking down an algebraic expression into simpler components. In the case of \(32-16x\), once we identify the GCF (16), we proceed to express the expression in its factored form. Here, the expression is transformed from a linear form to a product, reflecting its underlying structure.

To achieve this:

• Divide each term by the GCF, 16, to simplify each component.
• The term 32 divided by 16 results in 2.
• The term \(-16x\) divided by 16 results in \(-x\).

Thus, the expression \(32-16x\) becomes \(16(2-x)\). This transformation reveals the factored form, which is advantageous for solving equations, simplifying expressions, and understanding mathematical properties. Double-checking by expanding can verify the accuracy of the factoring process.

Algebraic Manipulation

Algebraic manipulation involves transforming expressions to demonstrate equivalencies or simplicity. It requires adeptness in mathematical operations and an understanding of properties of numbers. When factoring expressions, like \(32-16x\), manipulations ensure that we accurately extract common factors.

The process typically involves several steps:

• Identifying all parts of the expression (constants, variables).
• Applying operations—like division to factor out the GCF.
• Reconstructing the expression to its simplest or most manageable form.

Algebraic manipulation facilitates the solving of equations, graphing of functions, and logical deductions in higher-level mathematics. Recognizing how different operations interplay is central for accurate and efficient manipulation of algebraic expressions. Through practice and conceptual understanding, this skill becomes invaluable in various mathematical contexts.