1. Prerequisites: Fundamental Concepts of Algebra
Chapter P
Prerequisites: Fundamental Concepts of Algebra
P.6: Rational Expressions

2. Objectives At the end of this session, you will be able to:
Specify the numbers that must be excluded from the domain of rational expressions.
Simplify rational expressions.
Multiply rational expressions.
Divide rational expressions.
Add and subtract rational expressions.
Simplify complex rational expressions.

3. Index Rational Expressions Simplifying Rational Expressions
Multiplying Rational Expressions
Dividing Rational Expressions
Adding and Subtracting Rational Expressions
Complex Rational Expressions
Summary

4. 1. Rational Expressions
In the previous sections, we learned about integers and polynomials. Integers and polynomials have similar properties for addition and multiplication as given below:
From the above, we can say that polynomials ‘behave like integers’.
Recall: Rational Numbers:
The numbers of the form m/n, where m and n are integers, n  0, are called rational numbers.
S. No.
Integers
Polynomials 1.
The sum (difference) of two integers is an integer.
The sum (difference) of two polynomials is a polynomial. 2.
The integer zero (0) is such that for any integer a, a + 0 = a.
The zero polynomial (denoted by 0) is such that for any polynomial p(x), p(x) + 0 = p(x). 3.
For any integer a, there is an integer –a such that a + (-a) = 0.
For any polynomial p(x), there is a polynomial -p(x) such that p(x) + [-p(x)] = 0. 4.
The product of any two integers is an integer.
The product of any two polynomials is a polynomial. 5.
The integer 1 is such that for any integer a, a . 1 = a.
The constant polynomial 1 is such that for any polynomial p(x), p(x) . 1 = p(x).

5. 1. Rational Expressions(Cont…)
We discussed that integers and polynomials are algebraically similar. We, therefore introduce the algebraic expressions which behave like rational numbers and are named as rational expressions. We can define rational expressions as:
Rational Expressions:
An expression of the form , where p(x) and q(x) are polynomials and q(x)  0, is called a rational expression.
We call p(x) the numerator and q(x) the denominator of the rational expression
Thus, a rational expression is nothing more than a fraction in which the numerator and the denominator are polynomials.
Every polynomial is a rational expression, for, if p(x) be any polynomial, then we can write p(x) as
Thus, every polynomial is a rational expression, but every rational expression need not be a polynomial.
Examples:
is a rational expression whose numerator is a linear polynomial and the denominator
is a quadratic polynomial.
is a rational expression whose numerator is a cubic polynomial and the
denominator is a quadratic polynomial.
is not a rational expression, since the numerator is not a polynomial.

6. 1. Rational Expressions (Cont…)
Domain of rational expressions:
A rational expression is a quotient of two polynomials. All the operations that can be performed on regular fractions can be also be performed on rational expressions.
Fractions can have whole numbers for the numerator and denominator, but the denominator of a fraction cannot be equal to zero as division by zero is undefined.
Now as rational expressions involve division and division by zero is undefined, so while dealing with rational expressions we must exclude all the values from the domain of a rational expression, that make the denominator zero.
The set of real numbers for which an algebraic expression is defined is the domain of the expression.
For example let us find all the numbers that must be excluded from the domain of the following rational expressions:
For the given expression, the denominator would equal 0 if x = 3. Therefore, we must exclude 3 from the domain of the rational expression.
These excluded numbers are written to the right of the rational expression.

7. 1. Rational Expressions (Cont…)
To find the values to be excluded from the domain of the given rational expression, we first need to factorize the denominator.
x2 + 3x – 4 = (x - 1)(x + 4)
Since 1 would make the first factor of the denominator zero, so we will exclude 1.
Since -4 would make the second factor of the denominator zero, thus, we will exclude -4 from the domain of the rational expression.
Therefore, the values to be excluded from the domain of the given rational expression are 1 and -4.
Let us first factor the denominator.
x2 – 25 = (x - 5)(x + 5)
As 5 would make the first factor of the denominator zero, so we will exclude 5 from the domain of the given expression. Similarly, since -5 would make the second factor of the denominator zero, so we shall exclude -5 from the domain of the given expression.

8. 2. Simplifying Rational Expressions
We simplify a rational expression just as we simplify fractions. A rational expression is simplified or is in its lowest terms, if its numerator and denominator have no common factors other than 1 or –1.
To simplify rational expressions, we must first find all common factors (constants, variables, or polynomials) in the numerator and the denominator. Thus, we must factor the numerator and the denominator first. Once the numerator and the denominator have been factored, we cross out any common factors. You must remember that it is only common factors that you can divide out.
We use the following steps simplify rational expressions:
Step 1: Factor the numerator and denominator completely.
Step 2: Divide both the numerator and denominator of the expression by the common factor.
Let us take few examples to illustrate the above steps:
Simplify: 1 1

9. 2. Simplifying Rational Expressions(Cont..)
(Step 1: Factor the numerator and denominator;
As the denominator is (x+3)(x-9), x -3 and x 9)
(Step 2: Divide the numerator and denominator by common factor (x+3))
(Rational expression in simplified form)
NOTE: Each term in the numerator must have a factor that cancels a common factor in the denominator. You can only cancel a factor of the entire numerator with a factor of the entire denominator.
(Step 1: Factor the numerator and the denominator; As the denominator is (x-5), thus x 5)
(Step 2: Divide the numerator and denominator by common factor (x-5))
NOTE: We cannot cancel out 5 - x and x – 5. They only differ by signs, in other words they are opposites of each other.  In that case, we can factor -1 out of one of these factors and rewrite it with opposite signs, as shown in the above example. 1 1
These numbers that are excluded from the domain of the original expression should also be excluded from the domain of the simplified expression. 1 1

10. 3. Multiplying Rational Expressions
The multiplication of rational expressions is similar to the multiplication of rational numbers.
If are two rational numbers, then we know that their product is given by
Similarly, the product of two rational expressions is given by
The procedure for multiplying rational expressions is as follows:
Step 1: Factor all the numerators and denominators completely.
Step 2: Simplify the rational expressions.
Step 3: Multiply the remaining factors in the numerator and multiply the remaining factors in the denominator.
Let us solve few examples to show the these steps:
Example 1: Multiply the rational expressions
Solution: Step 1: Factor all the numerators and denominators completely.
(As the denominator is x(x-3)(x+6)(x+3), thus x 0, 3, -6, -3 )

11. 3. Multiplying Rational Expressions(Cont…)
Step 2: Simplify the rational expression.
(Rational expression in step 1)
(Simplify by dividing the numerator and denominator by the common factors)
NOTE: Any factor in the numerator can cancel with any factor in the denominator.
Step 3: Multiply the remaining factors in the numerator and multiply the remaining factors in the denominator.
NOTE: Even though all of the factors in the numerator were cancelled, a 1 still remains. It is easy to think there is “nothing” left and make the numerator disappear.  But when you divide a factor by itself, a 1 actually remains, just as 2/2 = 1 or 5/5 = 1. Also note that the values that would be excluded from the domain are 0, 3, -6, and -3.  These are the values that make the original denominator equal to 0. 1 1 1 1 1 1

12. 4. Dividing Rational Expressions
For each non-zero rational expression ,[p(x)0], there exists a rational expression such that
is called the reciprocal (or multiplicative inverse) of
To find the quotient of two rational expressions, we multiply the expression in the numerator by the reciprocal of the expression in the denominator.
The procedure for dividing rational expressions is as follows:
Step 1: Multiply the numerator by the reciprocal of the divisor.
Step 2: Multiply the rational expressions.
NOTE: Do not attempt to cancel factors before division is written as a multiplication.
Example: Divide and simplify
Reciprocal of the divisor

13. 4. Dividing Rational Expressions(Cont…)
Step 2: Multiply the rational expressions.
(Factor the numerator and denominator completely)
(Simplify the rational expression by dividing the numerator and denominator
by the common factors)
(Multiplying the remaining factors in the numerator and denominator)
NOTE: The values that would be excluded from the domain are -6 and 0. These are the values that make the original denominator of the product equal to 0.

14. 5. Adding and Subtracting Rational Expressions
Adding and Subtracting Rational Expressions with Common Denominators:
We add or subtract rational expressions the same way as we add or subtract rational numbers. We follow the following steps for adding or subtracting rational expressions with same denominators:
Step 1: Add or subtract the numerators.
Step 2: Put the sum or difference found in Step 1 over the common denominator.
Step 3: Simplify, if needed.
Now let us solve some examples showing these steps:
Example: Subtract
(Step 1: Subtracting the numerators and
Step 2: Placing this difference over the
common denominator)
(Changing the sign of each term in the second bracket)
(Combining like terms)
(Step 3: Simplify, x -1)

15. 5. Adding and Subtracting Rational Expressions(Cont…)
Adding and Subtracting Rational Expressions with Different Denominators:
We know that fractions can be added or subtracted only if they have the same denominator. Similarly, rational expressions can be added or subtracted only if they have the same denominator. Thus, to add or subtract two rational expressions with unlike denominators, we must rewrite them as expressions with a common denominator.
To add or subtract rational expressions with different denominators we find the least common denominator(LCD).
The LCD is the list of all the different factors in the denominators raised to the highest power that there is of each factor. In other words, the least common denominator is a polynomial consisting of the product of all prime factors in the denominators, with each factor raised to the greatest power of its occurrence in any denominator.
Finding the LCD:
Step 1: Factor each denominator completely.
Step 2: List the factors of the first denominator.
Step 3: Add to this list in step 2 any factors of the second denominator which are not there in the list.
Step 4: Find the product of all the factors in the final list.

16. 5. Adding and Subtracting Rational Expressions(Cont…)
For example let us find the LCD of
Step 1: Factor each denominator completely.
x2 - 6x + 9 = (x - 3)2
x2 – 9 = (x-3)(x+3)
Step 2: List the factors of the first denominator.
First denominator: x2 - 6x + 9 = (x - 3)2
So the list of factors of the first denominator: (x-3), (x-3)
Step 3: Add to this list in step 2 any factors of the second denominator which are not there in the list.
Second denominator: x2 – 9 = (x-3)(x+3)
So the factors of the second denominator are (x-3)(x+3).
Now one factor (x - 3) is already in the list but the second factor (x + 3) is not there in the list. So we add this second factor to the above list. Thus, our final list is
(x - 3), (x - 3), (x + 3)
Step 4: Find the product of all the factors in the final list.
(x - 3)(x - 3)(x + 3) = (x - 3)2 (x + 3)
Thus, (x - 3)2 (x + 3) is the required LCD of the given rational expressions.

17. Adding and Subtracting Rational Expressions (Cont…)
Next, for adding or subtracting rational expressions with different denominators we have the following steps:
Step 1: Find the LCD.
Step 2: Write all rational expressions in terms of the LCD.
For this multiply both the numerator and the denominator of each rational expression by any factor (s) needed to convert the original denominator of this expression into the LCD.
Step 3: Now we have the rational expressions with the same denominators, so we can add or subtract numerators as before.
Step 4: Simplify the resulting expression, if necessary.
Example: Subtract
Solution: Step 1: Find the LCD.
Factoring denominators: x2 + 6x + 5 = (x + 5)(x + 1)
x2 - 7x – 8 = (x - 8)(x + 1)
Factors of the first denominator are (x + 5)(x + 1).
Factors of the second denominator are (x - 8)(x + 1).
LCD is (x + 5)(x + 1)(x - 8).

18. 5. Adding and Subtracting Rational Expressions(Cont…)
Step 2: Write all rational expressions in terms of the LCD.
Rewriting the first expression with the LCD.
(Multiplying the numerator and denominator by the extra factor to form the LCD)
Rewriting the second expression with the LCD.
Thus, we have

19. 5. Adding and Subtracting Rational Expressions(Cont…)
Step 3: Now we have the rational expressions with the same denominators, so we can subtract numerators by placing this difference over the LCD.
(Removing the parentheses and changing the sign of each term in the second bracket)
(Combining like terms)
Step 4: Simplify the resulting expression, if necessary.
(Factor the numerator, no common factors to cancel out)

20. 6. Complex Rational Expressions
Complex rational expressions, also called complex fractions, are rational expressions which have a fraction in their numerator or their denominator or in both. In other words, there is at least one small fraction within the overall fraction.
Some examples of complex fractions are:
There are two methods to solve complex rational expressions:
Method 1: We follow the following steps for solving complex fractions by this method:
Step 1: If required, rewrite the numerator and denominator as simple fractions.  
In other words, we will combine all the parts of the numerator to form one fraction and all of the parts of the denominator to form another fraction. 
Step 2:  Divide the numerator by the denominator.   
When we find the quotient of two rational expressions, we multiply the numerator by the reciprocal of the denominator. So in this step we will multiply the numerator by the reciprocal of the denominator.
Step 3: If needed, simplify the rational expression.    

21. 6. Complex Rational Expressions (Cont…)
Let us solve an example to illustrate the steps for method 1:
Example 1: Simplify
Step 1: If required, rewrite the numerator and denominator as simple fractions.
Combining the numerator:
(Taking the LCD of the denominator)
(Rewriting the fractions with LCD ab)
Combining the denominator:
(Rewriting the fractions with LCD a2b)
Putting the complex fraction together:

22. 6. Complex Rational Expressions (Cont…)
Step 2: Divide the numerator by the denominator. 
(Rewriting division as multiplication of the reciprocal)
Step 3: Simplify, if required.
(Canceling out the common factors)
Note that the value to be excluded from the domain is 0. This is the value that makes the original denominator equal to zero.

23. 6. Complex Rational Expressions (Cont…)
Method 2: We follow the following steps for solving complex fractions by this method:
Step 1: Multiply the numerator and denominator of the overall complex fractions by the LCD of the smaller fractions. Recall that when we multiply the numerator and the denominator with the exact same thing, the result is an equivalent fraction.
Step 2: If needed, simplify the rational expression.
  Let us solve an example to illustrate the steps for method 2:
Example:
Step 1: Multiply the numerator and denominator of the overall complex fractions by the LCD of the smaller fractions.
Denominator of the numerator’s fraction has the following factor:
(x + 1)
Denominator of the denominator’s fraction has the following factor:
(x - 1)
LCD of the fractions: (x + 1)(x - 1)

24. 6. Complex Rational Expressions (Cont…)
Multiplying the numerator and denominator by the LCD, we get,
(Multiplying the numerator and denominator by (x + 1)(x - 1))
(Using distributive property)
(Canceling the common factors)
(Simplifying)

25. 6. Complex Rational Expressions (Cont…)
Step 2: Simplify the rational expression, if required.
Note that the values that would be excluded from the domain are -1, 1, and 0. These are the values that make the original denominator equal to 0.  

26. 7. Summary Let us recall what we have learnt so far:
Rational Expressions: A rational expression is a quotient of two polynomials.
A rational expression is simplified or is in its lowest terms, if its numerator and denominator have no common factors other than 1 or –1.
In this section we have studied the following operations on rational expressions:
Addition
Subtraction
Multiplication
Division