# Simplifying Rational Expressions: A Comprehensive Guide Unlock the secrets of simplifying rational expressions with our step-by-step approach. Learn to factor, identify restrictions, and cancel common terms. Elevate your algebra skills and tackle complex problems with confidence.

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Now Playing:Simplify rational expressions and restrictions– Example 0
Intros
1. Why is it important to determine the non-permissible values prior to simplifying a rational expression?
Examples
1. For each rational expression:
i) determine the non-permissible values of the variable, then
ii) simplify the rational expression
1. $\frac{{6{x^3}}}{{4x}}$

2. $\frac{{5 - {x}}}{{{x^2} - 8x + 15}}$

3. $\frac{{{x^2} + 13x + 40}}{{{x^2} - 25}}$

Simplifying rational expressions and restrictions
Notes
A rational expression is a fraction that its numerator and/or denominator are polynomials. In this lesson, we will first learn how to find the non-permissible values of the variable in a rational expression. Then, we will how to simplify rational expressions.
$\cdot$ multiplication rule: $x^a \cdot x^b=x^{a+b}$

$\cdot$ division rule: $\frac{x^a}{x^b}=x^{a-b}$
Concept

## Introduction to Rational Expressions

Algebraic fractions are algebraic fractions that consist of polynomials in both the numerator and denominator. Understanding these expressions is crucial in advanced algebra and calculus. Our introduction video provides a comprehensive overview of rational expressions, laying the foundation for more complex concepts. This lesson focuses on two key aspects: finding non-permissible values and simplifying rational expressions. Non-permissible values are critical as they represent values that make the denominator zero, leading to undefined expressions. Simplifying rational expressions involves reducing them to their simplest form by canceling common factors. This process not only makes calculations easier but also helps in identifying the expression's behavior. By mastering these skills, you'll be better equipped to handle more advanced mathematical problems involving rational expressions. Throughout this lesson, we'll provide step-by-step guidance and practice problems to reinforce your understanding of these fundamental concepts.

Example

For each rational expression:
i) determine the non-permissible values of the variable, then
ii) simplify the rational expression

$\frac{{6{x^3}}}{{4x}}$

#### Step 1: Determine the Non-Permissible Values

To determine the non-permissible values of the variable, we need to identify the values that would make the denominator equal to zero. In a rational expression, the denominator cannot be zero because division by zero is undefined. For the given expression $\frac{6x^3}{4x}$, the denominator is $4x$.

We set the denominator equal to zero and solve for $x$:

$4x = 0$

Solving for $x$, we get:

$x = 0$

Therefore, the non-permissible value for the variable $x$ is $0$. This means that $x$ cannot be $0$ in the given rational expression.

#### Step 2: Simplify the Rational Expression

Next, we simplify the rational expression $\frac{6x^3}{4x}$. To do this, we need to divide both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 6 and 4 is 2.

We start by dividing the coefficients:

$\frac{6}{4} = \frac{6 \div 2}{4 \div 2} = \frac{3}{2}$

Now, we simplify the variable part. We have $x^3$ in the numerator and $x$ in the denominator. When we divide $x^3$ by $x$, we subtract the exponents:

$\frac{x^3}{x} = x^{3-1} = x^2$

Combining the simplified coefficients and variables, we get:

$\frac{6x^3}{4x} = \frac{3x^2}{2}$

Therefore, the simplified form of the rational expression is $\frac{3x^2}{2}$.

FAQs

Here are some frequently asked questions about simplifying rational expressions:

#### 1. How do you simplify rational expressions step by step?

To simplify rational expressions, follow these steps: 1. Factor the numerator and denominator completely. 2. Identify common factors in the numerator and denominator. 3. Cancel out common factors. 4. Write the simplified expression. 5. State any restrictions on the variable.

#### 2. What is the first step in simplifying a rational expression?

The first step in simplifying a rational expression is to factor both the numerator and denominator completely. This allows you to identify common factors that can be canceled out.

#### 3. How do you evaluate and simplify rational expressions?

To evaluate and simplify rational expressions: 1. Determine non-permissible values by setting the denominator to zero. 2. Factor the numerator and denominator. 3. Cancel common factors. 4. Simplify any remaining numerical fractions. 5. State the domain restrictions.

#### 4. What are the 3 steps on simplifying rational expression?

The three main steps are: 1. Factor the numerator and denominator. 2. Identify and cancel common factors. 3. Write the simplified expression with domain restrictions.

#### 5. Can a rational function be simplified?

Yes, rational functions can be simplified by factoring the numerator and denominator, canceling common factors, and stating domain restrictions. However, not all rational functions will simplify to a simpler form.

Prerequisites

Understanding the foundation of simplifying rational expressions and restrictions is crucial for mastering this important algebraic concept. To excel in this area, students must first grasp several key prerequisite topics that form the building blocks of this skill.

One of the fundamental prerequisites is common factors of polynomials. This skill is essential because simplifying rational expressions often involves factoring both the numerator and denominator to identify and cancel common terms. Similarly, factoring polynomials is a critical skill that allows students to break down complex expressions into simpler forms.

Another crucial concept is understanding the domain and range of a function. This knowledge is particularly important when dealing with restrictions in rational expressions, as it helps identify values that would make the denominator zero, leading to undefined expressions. Related to this is the ability to determine non-permissible values for trig expressions, which applies similar principles to trigonometric functions.

The order of operations (PEMDAS) is a fundamental concept that ensures students simplify expressions correctly, maintaining the integrity of the mathematical operations. This is particularly important when dealing with complex rational expressions that involve multiple operations.

Understanding vertical asymptotes is also relevant, as it relates to the behavior of rational functions at certain x-values, which is crucial when analyzing the restrictions and domain of these functions. This concept ties in closely with graphing rational functions, another important skill that helps visualize the behavior of simplified rational expressions.

Lastly, proficiency in solving rational equations is a natural extension of simplifying rational expressions. This skill allows students to apply their knowledge of simplification to solve more complex problems involving rational expressions.

By mastering these prerequisite topics, students build a strong foundation for simplifying rational expressions and understanding their restrictions. Each concept contributes to a deeper comprehension of the subject, enabling students to approach more advanced problems with confidence and skill. Remember, mathematics is a cumulative subject, and a solid grasp of these fundamentals is key to success in more complex algebraic concepts.