# Prerequisites

## Rational Expressions


### Learning Objectives

1. Identify the skills leading to successful preparation for a college level mathematics exam.

2. Create a plan for success when taking mathematics exams.

### Objective 1: Identify the skills leading to successful preparation for a college level mathematics exam.

Complete the following surveys by placing a checkmark in the a column for each strategy based on the frequency that you engaged in the strategy during your last academic term.


### Practice Makes Perfect

Practice: Identify the skills leading to successful preparation for a college level mathematics exam.


### Objective 2: Create a plan for success when taking mathematics exams.

1. It’s important to take the opportunity to reflect on your past experiences in taking math exams as you begin a new term. We can learn a lot from these reflections and thus work toward developing a strategy for improvement.
In the table below list 5 challenges you have had in past math courses when taking an exam and list a possible solution that you could try this semester.


2. Develop your plan for success.  Keep in mind the idea of mindsets and try to approach your test taking strategies with a growth mindset. Now is the time for growth as you begin a new term. Share your plan with your study group members.

A pastry shop has fixed costs of 
  per week and variable costs of 
  per box of pastries. The shop’s costs per week in terms of 
  the number of boxes made, is 
  We can divide the costs per week by the number of boxes made to determine the cost per box of pastries.
Notice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will explore quotients of polynomial expressions.

### Simplifying Rational Expressions

The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let’s start with the rational expression shown.
We can factor the numerator and denominator to rewrite the expression.
Then we can simplify that expression by canceling the common factor 
 


### Multiplying Rational Expressions

Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.

### Dividing Rational Expressions

Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would rewrite 
  as the product 
  Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.

### Adding and Subtracting Rational Expressions

Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let’s look at an example of fraction addition.
We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.
The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were 
  and 
  then the LCD would be 
 

Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of 
  by 
  and the expression with a denominator of 
  by 
 


### Simplifying Complex Rational Expressions

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression 
  can be simplified by rewriting the numerator as the fraction 
  and combining the expressions in the denominator as 
  We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get 
  which is equal to 
 


### Key Concepts

1. Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See .
2. We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See .
3. To divide rational expressions, multiply by the reciprocal of the second expression. See .
4. Adding or subtracting rational expressions requires finding a common denominator. See  and .
5. Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, simplify the rational expressions.
For the following exercises, multiply the rational expressions and express the product in simplest form.
For the following exercises, divide the rational expressions.
For the following exercises, add and subtract the rational expressions, and then simplify.
For the following exercises, simplify the rational expression.

### Real-World Applications


### Extensions

For the following exercises, perform the given operations and simplify.

### Chapter Review Exercises


### Real Numbers: Algebra Essentials

For the following exercises, perform the given operations.
For the following exercises, solve the equation.
For the following exercises, simplify the expression.
For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

### Exponents and Scientific Notation

For the following exercises, simplify the expression.

### Radicals and Rational Expressions

For the following exercises, find the principal square root.

### Polynomials

For the following exercises, perform the given operations and simplify.

### Factoring Polynomials

For the following exercises, find the greatest common factor.
For the following exercises, factor the polynomial.

### Rational Expressions

For the following exercises, simplify the expression.

### Chapter Practice Test

For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.
For the following exercises, evaluate the expression.
For the following exercises, simplify the expression.
For the following exercises, factor the polynomial.
For the following exercises, simplify the expression.