Problem 6 - Algebra and Trigonometry

Question

\(5-12\) . Factor out the common factor. $$ -3 b+12 $$

Step-by-Step Solution

Answer

Factorization is \(3(-b + 4)\).

Step 1: Identify the Common Factor

Look at the coefficients and terms of the expression, \(-3b + 12\). The common factor in both terms is \(3\), as \(-3\) and \(12\) are both divisible by \(3\).

Step 2: Factor Out the Common Factor

Factor \(3\) out of each term. Divide each term by \(3\): \(-3b \div 3 = -b\) and \(12 \div 3 = 4\).The expression becomes \(3(-b + 4)\).

Step 3: Check the Factorization

Distribute the common factor back into the expression to ensure accuracy: \(3(-b + 4) = 3(-b) + 3(4) = -3b + 12\). Since the result matches the original expression, the factorization is correct.

Key Concepts

Common Factor

When working with algebraic expressions, one of the simplest yet most powerful techniques is factoring out the common factor. In essence, a common factor is a number or variable that is shared by all terms within an expression. This is important because it helps simplify complex expressions and solve equations more easily.
For instance, in the expression \(-3b + 12\), we look for a number that divides both \(-3b\) and \(12\). Here, the number is \(3\), as both \(-3\) and \(12\) can be divided evenly by \(3\).

• Understanding what a common factor is involves looking at the coefficients (the numbers in front of variables) and identifying the largest number that can divide each one without a remainder.
• This is useful in simplifying expressions and making calculations more manageable.

By factoring out the common factor, we transform the original expression into an equivalent but simpler form.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators such as addition, subtraction, multiplication, and division. They are used extensively in algebra to describe relationships, patterns, and real-world situations.
In the exercise we're analyzing, the expression \(-3b + 12\) is an algebraic expression. Breaking it down:

• \(-3b\) is a term consisting of a coefficient \(-3\) and a variable \(b\).
• \(12\) is a constant term, which is just a number without a variable.

An understanding of algebraic expressions involves recognizing each part and knowing how they come together to form the whole. This understanding is crucial when working with expressions, especially when performing operations like finding common factors.

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a single term across terms inside a parenthesis. It is expressed as \(a(b + c) = ab + ac\). This property can be applied in reverse when factoring out a common factor.
When we factor the expression \(-3b + 12\), we use the distributive property to "factor out" \(3\), leading to the expression \(3(-b + 4)\). Here’s how this works:

• You take the common factor \(3\) and place it outside the parenthesis.
• Then you divide each term inside the expression by \(3\), resulting in a simplified expression \((-b + 4)\) inside the parenthesis.

To verify the factorization, you can distribute the \(3\) back into \((-b + 4)\) to ensure you reach the original expression: \(3(-b + 4) = -3b + 12\). This check confirms that the process was done correctly.