Problem 74 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{4}{3 b^{2}-12 b}-\frac{2}{6 b^{2}-6 b} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: $$\frac{4}{3b^2 - 12b} - \frac{2}{6b^2 - 6b}$$ Answer: $$\frac{1}{(b-4)(b-1)}$$

Step 1: 1. Identify the Common Denominator

First, let's identify the common denominator. The two expressions have the following denominators: $$ 3b^2 - 12b \quad \text{and} \quad 6b^2 - 6b $$ To find the common denominator, we will need to factor out any common factors in each denominator.

Step 2: 2. Factor the Denominators

Let's factor the denominators: $$ 3b^2 - 12b = 3b(b - 4) $$ and $$ 6b^2 - 6b = 6b(b - 1) $$ Now the common denominator can be found by multiplying the unique factors from each denominator. So the common denominator is: $$ 3b \cdot 6b \cdot (b - 4) \cdot (b - 1) $$

Step 3: 3. Make Equivalent Fractions with the Common Denominator

Now that we have found the common denominator, let's rewrite the original expressions with the common denominator. First expression: $$ \frac{4}{3b^2 - 12b} = \frac{4}{3b(b - 4)} \cdot \frac{6b(b-1)}{6b(b-1)} = \frac{24b(b-1)}{18b^2(b-4)(b-1)} $$ Second expression: $$ \frac{2}{6b^2 - 6b} = \frac{2}{6b(b - 1)} \cdot \frac{3b(b-4)}{3b(b-4)} = \frac{6b(b-4)}{18b^2(b-4)(b-1)} $$

Step 4: 4. Add/Subtract the Fractions

Now we can subtract the two fractions since they have the same denominator: $$ \frac{24b(b - 1)}{18b^2(b - 4)(b - 1)} - \frac{6b(b - 4)}{18b^2(b - 4)(b - 1)} $$ Combine the numerators: $$ \frac{24b(b-1) - 6b(b-4)}{18b^2(b-4)(b-1)} $$

Step 5: 5. Simplify the Numerator

Expand the numerators and simplify: $$ \frac{24b^2 - 24b - 6b^2 + 24b}{18b^2(b-4)(b-1)} = \frac{18b^2}{18b^2(b-4)(b-1)} $$

Step 6: 6. Simplify the Expression

Now, cancel out the common factors in the numerator and the denominator: $$ \frac{18b^2}{18b^2(b-4)(b-1)} = \frac{1}{(b-4)(b-1)} $$ So, the simplified expression is: $$ \frac{1}{(b-4)(b-1)} $$

Key Concepts

Common Denominator

When adding or subtracting rational expressions, finding a common denominator is vital. Just like when you add or subtract fractions with different denominators, you must find a common base you can use to combine them. This is often one of the main hurdles when working with algebraic fractions, as unlike with numerical fractions, you're dealing with variables and possibly more complex factors.

For example, consider the denominators in our exercise \( 3b^2 - 12b \) and \( 6b^2 - 6b \) which at first glance may not seem to have much in common. However, by factoring them, we can notice shared factors that will allow us to identify the least common denominator. This step is essential because it lays the foundation for correctly combining the rational expressions.

Factorization

Factorization is the process of breaking down an expression into a product of more simple expressions—its factors. These factors can be numbers, variables, or both, and are generally easier to work with than the original expression. In our exercise, \( 3b^2 - 12b \) was factored into \( 3b(b - 4) \) and \( 6b^2 - 6b \) into \( 6b(b - 1) \.

This is critical because it reveals the common factors between the denominators and therefore simplifies finding the common denominator. Understanding how to factor expressions correctly helps to drastically simplify the process of adding or subtracting rational expressions, as well as solve various other algebra problems.

Equivalent Fractions

In our search to combine rational expressions, equivalent fractions come into play. They are different fractions that represent the same value or proportion. When dealing with algebraic expressions, we create equivalent fractions by multiplying the numerator and denominator by the same expression, which is essentially multiplying by one.

This was done in the problem by multiplying the first expression's numerator and denominator by \( 6b(b-1) \) and the second by \( 3b(b-4) \). The goal is for both original fractions to have the same denominator, thus transforming them into equivalent fractions with a shared base. This step is key for being able to add or subtract them.

Simplify Rational Expressions

Once the rational expressions have been combined, the resulting expression often needs simplification. Simplifying may involve expanding the numerators and denominators, combining like terms, and canceling out common factors. This makes the expression more straightforward and neat.

In the solution, we simplified by expanding and then canceling out the common factors. This left us with \( \frac{1}{(b-4)(b-1)} \), a much simpler and more elegant expression. Simplifying not only makes the expressions easier to read but also, in many cases, is necessary to find the correct answer. Mastering simplification can lead to a deeper understanding of algebra and a better ability to solve complex problems.