Simplifying Algebraic Expressions

Let’s take our understanding of algebraic fractions to the next level! This post will address more complex problems that involve both removing parentheses and working with expressions within the numerator and denominator.

General Rules for Simplifying Algebraic Expressions

Combine Like Terms: Like terms are terms that have the same variables and powers. The coefficients of these terms can be added or subtracted.
- Example: In the expression $4x + 3x$ , $4x$ and $3x$ are like terms and can be combined to give $7x$ .
Use the Distributive Property: The distributive property states that . This property is used when you have a term being multiplied by a bracket.
- Example: In the expression $2(3x + 4)$ , you can distribute the $2$ to get $6x + 8$ .
Apply the Order of Operations: The order of operations (PEMDAS/BODMAS) is Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Always perform operations in this order.
- Example: In the expression $3 + 2 \times (4 - 2)^2$ , first solve the operation in the parentheses $(4 - 2)$ , then square the result, multiply by $2$ and finally add $3$ . The simplified result is $11$ .
Simplify Fractions: When dealing with fractions, always simplify them to their lowest terms. This can be done by dividing the numerator and the denominator by their greatest common factor.
- Example: The fraction $\frac{8}{12}$ can be simplified to $\frac{2}{3}$ by dividing both the numerator and the denominator by their greatest common factor, which is $4$ .
Factor When Possible: Factoring is the process of breaking down an expression into its simplest factors. It can simplify complex expressions and make them easier to work with.
- Example: The expression $x^2 - 4x + 4$ can be factored to $(x - 2)^2$ .

Simplifying Algebraic Fractions

Let’s start with a simple example. Suppose we are asked to simplify the following expression:

$\frac{a + 4}{3} - \frac{a}{3}$

Here, the least common multiple (LCM) of the denominators is 3. Therefore, we can subtract the fractions directly:

$\frac{a + 4 - a}{3} = \frac{4}{3}$

This can be further simplified to $1\frac{1}{3}$ .

Practical Examples

Download Student Notes To View Step By Guides For These Examples

Let’s consider a more complex example

Example 1:

$\frac{2c - d}{3} + \frac{3c + 2d}{6}$

Here, the LCM of the denominators is 6. Therefore, we can rewrite the expression as:

$\frac{2(2c - d) + (3c + 2d)}{6} = \frac{4c - 2d + 3c + 2d}{6} = \frac{7c}{6}$

Example 2:

$\frac{e - 3f}{4} - \frac{2e + f}{3}$

Here, the LCM of the denominators ( $4$ and $3$ ) is $12$ . Therefore, we can rewrite the expression as:

$\frac{3(e - 3f) - 4(2e + f)}{12} = \frac{3e - 9f - 8e - 4f}{12} = \frac{-5e - 13f}{12}$

Dealing with Expressions in the Denominator

Now, let’s consider cases where the denominators are expressions.

Example 3:

$\frac{1}{x} + \frac{2}{x + 1}$

Here, we treat $x$ and $x + 1$ as prime numbers. Therefore, the LCM of the denominators is $x(x + 1)$ . We can rewrite the expression as:

$\frac{1(x + 1) + 2x}{x(x + 1)} = \frac{3x + 1}{x(x + 1)}$

Example 4:

$\frac{x}{x + 2} + \frac{4}{x + 3}$

To simplify the expression

$\frac{x}{x + 2} + \frac{4}{x + 3}$

we first need to find a common denominator. In this case, the least common denominator (common multiple) is $(x + 2)(x + 3)$ .

Next, we rewrite each fraction with the common denominator and simplify:

$\frac{x(x + 3)}{(x + 2)(x + 3)} + \frac{4(x + 2)}{(x + 2)(x + 3)}$

This simplifies to:

$\frac{x^2 + 3x + 4x + 8}{(x + 2)(x + 3)}$

Combine like terms in the numerator:

$\frac{x^2 + 7x + 8}{(x + 2)(x + 3)}$

So, the simplified form of the given expression is

$\frac{x^2 + 7x + 8}{(x + 2)(x + 3)}$

Conclusion

In conclusion, simplifying algebraic expressions involves finding the least common multiple of the denominators and using it to combine the fractions. This process can be applied even when the denominators are expressions. In the next lesson, we will apply these concepts to solve linear equations.

Student Notes