Problem 15 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{12 b-6}{3} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is \(4b - 2\).

Step 1: Identify the common factor

In this case, the expression is \(\frac{12b-6}{3}\). We can observe that both the terms in the numerator (12b and 6) have a common factor of 3, which is also the denominator.

Step 2: Distribute the common factor

Now, we can distribute the common factor (3) to both terms in the numerator. So we have: $$ \frac{3(4b - 2)}{3} $$

Step 3: Cancel the common factor

Since we have a common factor in the numerator and denominator, we can cancel it out: $$ \frac{3(4b - 2)}{3} = 4b - 2 $$ So, the simplified expression after the division is \(4b - 2\).

Key Concepts

Understanding Common Factors in Algebra

In algebra, simplifying expressions often begins with identifying common factors. A common factor is a number or algebraic term that divides into two or more other terms without leaving a remainder. For instance, when looking at the expression \(\frac{12b-6}{3}\), it's important to notice that both 12b and -6 have a common factor of 3. This commonality means we can simplify the expression by dividing each term by this factor.

Recognizing common factors ensures that algebraic expressions are as simple as possible. This not only makes expressions easier to understand but also sets the stage for more complex algebraic manipulation such as solving equations or factoring polynomials. When we use common factors efficiently, we streamline our work, prevent errors, and better grasp the relationships between algebraic terms.

The Process of Algebraic Division

Algebraic division is similar to numerical division but involves variables as well as numbers. It aims to simplify expressions or solve equations by dividing both the numerator and the denominator by the same nonzero number or term. As an example, let's revisit the exercise with the expression \(\frac{12b-6}{3}\).

We applied algebraic division by recognizing that each term in the numerator could be evenly divided by 3. In this context, we use the common factor to reduce the expression to its simplest form, \(4b - 2\), as seen in the solution. This technique is crucial to simplifying fractions and solving algebraic expressions. Understanding how to divide terms correctly is a foundational skill in algebra that will be used in various mathematical scenarios.

Applying the Distributive Property

The distributive property is a staple in algebra that allows us to multiply a single term by each term within a parenthesis. It provides a way to simplify and solve expressions efficiently. Following our textbook exercise, once the common factor of 3 is identified, it is 'distributed' across the terms 4b and -2 within the parentheses: \(3\times4b\) and \(3\times(-2)\), which is rewritten as \(3(4b - 2)\).

This property reminds us that multiplication over addition or subtraction can be 'distributed' across all terms. After applying the distributive property, we can cancel out the common factor in both the numerator and the denominator, which is a crucial step in simplifying algebraic fractions. Mastering the distributive property makes it simpler to work with algebraic expressions and prepares students for more complex operations such as expanding brackets and multiplying polynomials.