1. Home
  2. >Math 30-2 (Alberta)
  3. >Rational Expressions
Help

Simplifying rational expressions and restrictions

0/1
?
Intros
Lessons
  1. Why is it important to determine the non-permissible values prior to simplifying a rational expression?
0/13
?
Examples
Lessons
  1. For each rational expression:
    i) determine the non-permissible values of the variable, then
    ii) simplify the rational expression
    1. 6x34x\frac{{6{x^3}}}{{4x}}
    2. 5−xx2−8x+15\frac{{5 - {x}}}{{{x^2} - 8x + 15}}
    3. x2+13x+40x2−25\frac{{{x^2} + 13x + 40}}{{{x^2} - 25}}
  2. For each rational expression:
    i) determine the non-permissible values of the variable, then
    ii) simplify the rational expression
    1. 9t3−16t3t2+4t\frac{{9{t^3} - 16t}}{{3{t^2} + 4t}}
    2. x2+2x−3x4−10x2+9\frac{{{x^2} + 2x - 3}}{{{x^4} - 10{x^2} + 9}}
  3. For each rational expression:
    i) determine the non-permissible values of the variable, then
    ii) simplify the rational expression
    1. x−33−x\frac{{x - 3}}{{3 - x}}
    2. 5y3−10y230−15y\frac{{5{y^3} - 10{y^2}}}{{30 - 15y}}
    3. 1−9x26x2−7x−3\frac{{1 - 9{x^2}}}{{6{x^2} - 7x - 3}}
  4. The area of a rectangular window can be expressed as 4x2+13x+34{x^2} + 13x + 3, while its length can be expressed as 4x+14x + 1.
    1. Find the width of the window.
    2. If the perimeter of the window is 68 mm, what is the value of xx?
    3. If a cleaning company charges $3/m2m^2 for cleaning the window, how much does it cost to clean the window?
  5. For each rational expression:
    i) determine the non-permissible values for yy in terms of xx , then
    ii) simplify, where possible.
    1. 2x+y2x−y\frac{{2x + y}}{{2x - y}}
    2. x−3yx2−9y2\frac{{x - 3y}}{{{x^2} - 9{y^2}}}
0/17
?
Practice
Topic 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.

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.

Understanding Rational Expressions

Rational expressions are a fundamental concept in algebra, representing fractions that contain polynomials in either the numerator, denominator, or both. These expressions play a crucial role in various mathematical applications and problem-solving scenarios. To fully grasp the concept of rational expressions, it's essential to understand their structure and the importance of restrictions.

A rational expression is defined as a fraction where both the numerator and denominator are polynomials. For example, (x² + 3x) / (2x - 1) is a rational expression. The numerator (x² + 3x) and the denominator (2x - 1) are both polynomials. It's important to note that a polynomial in the numerator alone, such as (x³ + 2x² - 5) / 3, is also considered a rational expression, as the denominator is simply a constant.

One of the most critical aspects of working with rational expressions is understanding and identifying restrictions. Restrictions are values that make the denominator of a rational expression equal to zero. These restrictions are crucial because division by zero is undefined in mathematics. For instance, in the expression (x + 2) / (x - 3), the restriction is x 3, as this value would make the denominator zero.

To illustrate this concept further, let's consider the rational expression (x² - 4) / (x + 2). At first glance, it might seem that there are no restrictions. However, upon closer inspection, we can factor the numerator to (x + 2)(x - 2) / (x + 2). This reveals that x = -2 is a restriction, as it would make both the numerator and denominator zero, resulting in an undefined expression.

Understanding restrictions is not only important for mathematical accuracy but also for real-world applications. In physics, engineering, and economics, rational expressions often represent ratios or rates that have physical meaning. Ignoring restrictions could lead to erroneous conclusions or impossible scenarios in these fields.

When simplifying rational expressions, it's crucial to keep track of any factors that are canceled between the numerator and denominator. These canceled factors can introduce extraneous solutions or hide restrictions. For example, simplifying (x² - 1) / (x - 1) to (x + 1) introduces the restriction x 1, which wasn't present in the original expression.

In conclusion, rational expressions are a powerful tool in algebra, representing complex relationships between polynomials. By understanding their structure and the critical role of restrictions, students and professionals can effectively work with these expressions in various mathematical and real-world contexts. Always remember: the denominator of a rational expression can never be zero, and identifying these restrictions is key to correctly interpreting and using these expressions.

Determining Non-Permissible Values

When working with rational expressions, one crucial step that often gets overlooked is finding the non-permissible values, also known as restrictions. These are the values that would make the denominator of the expression equal to zero, which is mathematically undefined. Understanding and determining these values is essential before simplifying or performing any operations on rational expressions.

Let's explore this concept using the example from the video, where we have a rational expression with (w-2) and (w+4) in the denominator. To find the non-permissible values, we need to solve for w in each factor of the denominator:

1. For (w-2):
w - 2 = 0
w = 2

2. For (w+4):
w + 4 = 0
w = -4

Therefore, the non-permissible values for this rational expression are w = 2 and w = -4. These values are critical because they represent the points at which the expression is undefined, as division by zero is not allowed in mathematics.

It's important to emphasize that determining these non-permissible values should be done before any simplification or manipulation of the rational expression. This step is crucial for several reasons:

1. Maintaining mathematical accuracy: By identifying the restrictions upfront, you ensure that your subsequent calculations and simplifications are valid for all permissible values of the variable.

2. Avoiding errors in domain: Non-permissible values define the domain of the rational function. Knowing these values helps you understand where the function is defined and where it isn't.

3. Preventing division by zero: Identifying these values safeguards against accidentally dividing by zero during simplification or solving processes.

4. Preparing for graphing: If you need to graph the rational function, knowing the non-permissible values helps you identify vertical asymptotes.

When simplifying rational expressions, always start by finding these restrictions. Write them down separately from your main work, as they will be needed throughout the problem-solving process. After determining the non-permissible values, you can proceed with other steps such as factoring, canceling common factors, or performing operations on rational expressions with confidence.

Remember, the process of finding non-permissible values is straightforward:

1. Identify all factors in the denominator.
2. Set each factor equal to zero.
3. Solve for the variable in each equation.
4. List all solutions as your non-permissible values.

By mastering this crucial step in handling rational expressions, you'll build a solid foundation for more complex operations and ensure the accuracy of your mathematical work. Always keep in mind that these restrictions play a vital role in defining the behavior and properties of rational functions, making them an indispensable part of your problem-solving toolkit.

Simplifying Rational Expressions

Simplifying rational expressions is a crucial skill in algebra that involves reducing fractions containing variables to their simplest form. This process is essential for solving complex mathematical problems and equations. In this section, we'll explore the step-by-step process of simplifying rationals and provide examples to illustrate the concept.

To begin simplifying rational expressions, we need to identify common factors in the numerator and denominator. These common factors can be numbers, variables, or combinations of both. Once identified, we can cancel them out, effectively reducing the expression to its simplest form.

Let's consider the example from the video to illustrate this process:

(x² + 5x + 6) / (x² + 7x + 12)

To simplify this expression, we follow these steps:

  1. Factor the numerator and denominator separately:
    Numerator: x² + 5x + 6 = (x + 2)(x + 3)
    Denominator: x² + 7x + 12 = (x + 3)(x + 4)
  2. Identify common factors: (x + 3) appears in both the numerator and denominator
  3. Cancel out the common factor:
    (x + 2)(x + 3) / (x + 3)(x + 4)
  4. Write the simplified expression: (x + 2) / (x + 4)

This process of simplifying rationals can be applied to various types of expressions. Here are additional examples:

Example 1: (3x² + 12x) / (9x)

  1. Factor out common factors:
    Numerator: 3x(x + 4)
    Denominator: 9x
  2. Identify and cancel common factors: 3x
    3x(x + 4) / 3x3
  3. Simplified expression: (x + 4) / 3

Example 2: (x² - 4) / (x - 2)

  1. Factor the numerator: (x + 2)(x - 2)
  2. Identify and cancel common factors: (x - 2)
    (x + 2)(x - 2) / (x - 2)
  3. Simplified expression: (x + 2)

When simplifying rational expressions, it's important to remember these key points:

  • Always factor the numerator and denominator completely before canceling
  • Only cancel factors, not terms
  • Be aware of restrictions on the variable to avoid division by zero
  • Simplify any remaining numerical fractions if possible

Mastering the art of simplifying rationals is essential for advancing in algebra and calculus. It allows for easier manipulation of complex expressions and helps in solving equations more efficiently. Practice with various types of rational expressions to improve your skills and become proficient in this fundamental mathematical technique.

Remember, the key to successfully simplifying rational expressions lies in identifying common factors and canceling them out. This process not only reduces the complexity of the expression but also helps in understanding the underlying mathematical relationships between the numerator and denominator. As you continue to work with more complex rational expressions, you'll find that this skill becomes increasingly valuable in solving a wide range of mathematical problems.

Common Mistakes in Simplifying Rational Expressions

When working with rational expressions, one of the most common mistakes students make is simplifying before determining non-permissible values. This error can lead to overlooking crucial restrictions and potentially arriving at incorrect solutions. To avoid this pitfall, it's essential to understand the correct order of operations when dealing with rational expressions.

Let's examine a specific example from the video to illustrate this point. Consider the rational expression:

(x² - 1) / (x - 1)

At first glance, many students might be tempted to simplify this expression immediately. They might recognize that (x² - 1) can be factored as (x + 1)(x - 1), leading to the simplified form:

(x + 1)(x - 1) / (x - 1) = x + 1

While this simplification is mathematically correct, it overlooks a crucial step in the process: determining non-permissible values. By simplifying first, we've inadvertently removed an important restriction on the domain of the expression.

The correct approach is to determine non-permissible values before simplifying. In this case, we need to consider what value of x would make the denominator equal to zero:

x - 1 = 0

x = 1

This tells us that x = 1 is a non-permissible value for the original expression. It's crucial to note this restriction before proceeding with any simplification.

Now, let's look at the correct order of operations for handling rational expressions:

  1. Determine non-permissible values by setting the denominator equal to zero and solving for x.
  2. Factor the numerator and denominator if possible.
  3. Identify and cancel common factors between the numerator and denominator.
  4. Simplify the resulting expression.

Following this order ensures that we don't lose sight of important domain restrictions. In our example, the final answer would be:

x + 1, where x 1

This notation clearly indicates both the simplified expression and the restriction on its domain.

The importance of this approach becomes even more apparent when dealing with more complex rational expressions. For instance, consider:

(x² - 4) / (x² - 2x)

If we simplify first, we might get:

(x + 2)(x - 2) / (x(x - 2)) = (x + 2) / x

However, this simplification fails to account for the fact that x = 2 is also a non-permissible value in the original expression. By determining non-permissible values first, we would find that both x = 0 and x = 2 are restrictions, leading to the correct solution:

(x + 2) / x, where x 0 and x 2

In conclusion, when working with rational expressions, always remember to determine non-permissible values before simplifying. This approach ensures that you don't overlook critical domain restrictions and helps you arrive at complete and accurate solutions. By following the correct order of operations, you'll develop a more thorough understanding of rational expressions and avoid common pitfalls in your mathematical problem-solving.

Practice Problems and Solutions

Mastering the art of simplifying rational expressions and identifying non-permissible values is crucial for students advancing in algebra. Let's dive into a set of practice problems that will help reinforce these important concepts.

Problem 1: Simplify the following rational expression

(x² + 5x + 6) / (x² + 7x + 12)

Solution:

  1. Factor the numerator and denominator: (x + 2)(x + 3) / (x + 3)(x + 4)
  2. Cancel common factors: (x + 2) / (x + 4)
  3. Non-permissible value: x -4

Problem 2: Simplify and find non-permissible values

(x² - 4) / (x - 2)

Solution:

  1. Factor the numerator: (x + 2)(x - 2) / (x - 2)
  2. Cancel common factors: (x + 2)
  3. Non-permissible value: x 2

Problem 3: Simplify the complex fraction

(1/x + 1/y) / (1/x - 1/y)

Solution:

  1. Find a common denominator in both numerator and denominator: ((y + x) / xy) / ((y - x) / xy)
  2. Multiply by the reciprocal: ((y + x) / xy) * (xy / (y - x))
  3. Cancel common factors: (y + x) / (y - x)
  4. Non-permissible values: x y, x 0, y 0

Problem 4: Simplify and state restrictions

(x² - 9) / (x² - 2x - 3)

Solution:

  1. Factor both numerator and denominator: (x + 3)(x - 3) / (x + 1)(x - 3)
  2. Cancel common factors: (x + 3) / (x + 1)
  3. Non-permissible value: x -1

Problem 5: Simplify the following expression

(x³ - 8) / (x² - 4)

Solution:

  1. Factor the numerator and denominator: (x - 2)(x² + 2x + 4) / (x + 2)(x - 2)
  2. Cancel common factors: (x² + 2x + 4) / (x + 2)
  3. Non-permissible values: x -2, x 2

These practice problems cover a range of scenarios in simplifying rational expressions. Remember to always follow these key steps:

  1. Factor both the numerator and denominator completely.
  2. Identify and cancel common factors.
  3. Determine non-permissible values by setting the denominator equal to zero.

By consistently applying these steps, you'll become proficient in handling rational expressions. Keep in mind that

Conclusion

In this lesson, we've explored the crucial process of simplifying rational expressions, with a particular focus on identifying non-permissible values. The introduction video provided a solid foundation for understanding these concepts, emphasizing their importance in algebraic manipulation. Remember, identifying non-permissible values is a critical first step before simplifying any rational expression, as it prevents potential errors and ensures mathematical accuracy. This skill is essential for solving complex problems involving rational expressions in future lessons. Regular practice is key to mastering these techniques, so don't hesitate to work through additional examples and exercises. If you encounter difficulties, don't be afraid to seek help from your instructor or peers. By consistently applying these principles, you'll develop a strong grasp of rational expressions techniques, which will serve you well in more advanced mathematical studies. Keep up the great work and continue to build your algebraic skills!

Example:

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

6x34x\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 6x34x\frac{6x^3}{4x}, the denominator is 4x4x.

We set the denominator equal to zero and solve for xx:

4x=04x = 0

Solving for xx, we get:

x=0x = 0

Therefore, the non-permissible value for the variable xx is 00. This means that xx cannot be 00 in the given rational expression.

Step 2: Simplify the Rational Expression

Next, we simplify the rational expression 6x34x\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:

64=6÷24÷2=32\frac{6}{4} = \frac{6 \div 2}{4 \div 2} = \frac{3}{2}

Now, we simplify the variable part. We have x3x^3 in the numerator and xx in the denominator. When we divide x3x^3 by xx, we subtract the exponents:

x3x=x3−1=x2\frac{x^3}{x} = x^{3-1} = x^2

Combining the simplified coefficients and variables, we get:

6x34x=3x22\frac{6x^3}{4x} = \frac{3x^2}{2}

Therefore, the simplified form of the rational expression is 3x22\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.

Prerequisite Topics

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.

⋅\cdot multiplication rule: xa⋅xb=xa+bx^a \cdot x^b=x^{a+b}

⋅\cdot division rule: xaxb=xa−b\frac{x^a}{x^b}=x^{a-b}
Basic Concepts
?
Related Concepts
?