Thinking Mathematically by Robert Blitzer

1

• P.6 Rational Expressions
• EX Find all the numbers that must be excluded
  from the domain of each rational expression.
• y b. x 1
• x-2 x2 - 1

Solution To determine the numbers that must be
excluded from each domain, examine the
denominators. You must _EXCLUDE_from the
DOMAIN any value for the variable(s) that would
make the _DENOMINATOR_ (before reducing)
___ZERO______________. a. b.
2
Simplifying Rational Expressions

1. Factor the numerator and denominator completely.
2. Reduce (Divide both the numerator and denominator
   by the common factors. Remember a factor is
   something that is being MULTIPLIED!)

3

• Ex Simplify

Solution (note the form on simplifying the
expression)
You CANNOT reduce by dividing out 2 as 2 is not a
FACTOR (multiplied) in both.
Q What is the domain? Ans
4
Multiplying Rational Expressions

1. FACTOR all numerators and denominators
   completely.
2. REDUCE (Divide both the numerator and denominator
   by common factors.)
3. MULTIPLY the remaining factors in the numerator
   and multiply the remaining factors in the
   denominator.

5

• EX Multiply and simplify

6
When we divide rational expressions, we keep,
change, flip so that our result is in the form
of a multiplication problem, so we can apply the
previous rules. Example

• EX Divide and simplify

7
To ADD ( or subtract) rational expressions, we do
the same thing we would for a fraction Add the
numerators, and keep the denominators exactly the
same. (Then reduce if possible).Example

• Add

Solution
8
Finding the Least Common Denominator

• Factor each denominator completely.
• List the factors of the first denominator.
• Add to the list in step 2 any factors of the
  second denominator that do not appear in the
  list.
• Form the product of each different factor from
  the list in step 3. This product is the least
  common denominator.
• (If needed will do a brief example.)

9
Adding and Subtracting Rational Expressions That
Have Different Denominators with Shared Factors

1. LCD Find the least common denominator.
2. EQUIVALENT EXPRESSIONS Write all rational
   expressions in terms of the LCD. To do so,
   multiply both the numerator and the denominator
   of each rational expression by any factor(s)
   needed to convert the denominator into the least
   common denominator.
3. ADD or subtract the numerators, placing the
   resulting expression over the least common
   denominator.
4. REDUCE If necessary, simplify the resulting
   rational expression.

10

• EX Subtract

Solution
LCD (factor and find) Equivalent expressions Add
numerator/(keep denominator same) Reduce if
possible
11
Simplifying COMPLEX rational expressions

• We will take out the fraction within a fraction
  by multiplying both the numerator AND denominator
  by the LCD of all terms in the numerator and
  denominator. (See ex 9 p 67 for alternative
  method.)
• Ex Simplify

LCD?
12
Dont hesitate to ask about this type of problem!