Problem 2 - Algebra and Trigonometry

Question

1–8 ? Factor out the common factor. $$ -3 b+12 $$

Step-by-Step Solution

Answer

The expression factors to \(-3(b - 4)\).

Step 1: Identify Common Factor

Look for the greatest common factor (GCF) in the terms of the expression \(-3b + 12\). In this case, both terms can be divided by \(3\), which is a common factor, and since the first term is negative, the common factor is \(-3\).

Step 2: Divide Each Term by the Common Factor

Divide each term in the expression by the common factor \(-3\). For \(-3b\), divide \(-3b\) by \(-3\) to get \(b\). For \(12\), divide \(12\) by \(-3\) to get \(-4\).

Step 3: Factor Out the Common Factor

Rewrite the expression by factoring out \(-3\), giving \(-3(b - 4)\). This expresses the original expression \(-3b + 12\) as a product of its common factor and the simplified expression in parentheses.

Key Concepts

Greatest Common Factor (GCF)

When working with algebraic expressions, one of the most useful techniques you can learn is factoring, especially finding the Greatest Common Factor, or GCF. The GCF is the highest number that can divide each term of an expression without leaving a remainder. Finding the GCF is a foundational step that helps to simplify expressions and solve equations.

• For example, in the expression \(-3b + 12\), we look for the highest number that divides both \(-3b\) and \(+12\).
• Here, the number \(-3\) is common to both. Since the first term is negative, we choose \(-3\) to keep a standard form of factoring, which involves bringing out the negative first if present.

Understanding the GCF is essential because it helps to remove any unnecessary complexity from an expression, making further operations easier. Always remember to consider both numbers and any variables that may appear in multiple terms when determining the GCF.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They can represent real-world quantities or simple mathematics. Learning to manipulate these expressions, including factoring, can help solve mathematical problems efficiently.
A typical algebraic expression like \(-3b + 12\) has:

• Terms: Individual parts of the expression, separated by addition or subtraction. In this case, \-3b\ and \+12\.
• Factors: Components that multiply to form each term. For \-3b\, the factors are \-3\ and \b\.
• Coefficients: Numbers multiplying a variable. Here, \-3\ is the coefficient of \b\.

Grasping these concepts is crucial because it helps identify how to approach the expression for factoring or other algebraic operations.

Simplification

Simplification is the process of reducing an expression to its most basic form. This makes further calculations easier and reveals underlying patterns or solutions. For the expression \(-3b + 12\), simplifying involves factoring out the common factor \(-3\).

• First, divide each term by \(-3\). For \(-3b\), dividing gives \b\; for \+12\, it gives \-4\.
• Next, express the simplified terms within a parenthesis, resulting in something like \(-3(b - 4)\).

Simplification is powerful because it often transforms a complex-looking expression into a form that's easier to understand and work with. By breaking it down, you can gain insights into the problem at hand, whether it’s arithmetic operations, solving for a variable, or preparing for more advanced mathematical procedures.