Chapter 11 – Rational Expressions
Chapter 11 – Rational Expressions
Lesson #3 – Simplifying Rational Expressions
OBJECTIVES:
- Factor the numerator and denominator to simplify rational expressions.
- State the restrictions on the variable of the simplified rational expression.
- Extend simplification techniques to other algebraic fractions.
________________________________________________________________________
Q: Which expression is easier to solve for x=5?
[pic] or [pic].
A: Although the two equations are equivalent, one is easier to solve at first glance (and without a calculator) than the other! Help students to recall how they would have rewritten [pic]. The same procedure is used to rewrite rational expressions into their simplest form.
Simplest Form: When the numerator and denominator have no common factors other than 1 and -1.
[pic]
Fundamental Principle of
Rational Expressions
For any rational expression [pic], and any polynomial R, where ,[pic] , then
[pic]
In other words, if you multiply the EXACT SAME thing to the numerator and denominator, then you have an equivalent rational expression. This will come in handy when we simplify rational expressions, which is coming up next.
Simplifying a Rational Expression
Step 1: Factor the numerator and the denominator.
Step 2: Express each common factor pair in the numerator and denominator as 1.
Step 3: Simplify and state any restrictions on the variable.
[pic]
Example One: Simplify and find all numbers that must be excluded from the domain of the simplified rational expression: [pic].
Step 1: Factor the numerator and the denominator.
AND
Step 2: Divide out all common factors the numerator and the denominator have.
[pic] *Factor numerator and denominator.
[pic] *Divide out common factor of (x+3).
[pic] *Rational expression simplified.
Step 3: Simplify and state any restrictions on the variable.
To find the value(s) needed to be excluded from the domain, we need to ask ourselves, what value(s) of x would cause our denominator to be 0? Looking at the denominator x - 9, I would say it would have to be x = 9. Don’t you agree? 9 would be our excluded value.
Follow-up Example: Simplify [pic], where a [pic]0.
[pic]
Pre-Homework Problems:
Q: For what values of the variable is each rational expression undefined?
#18 [pic] (x[pic]3) #23 [pic](x[pic]3)
Q: Name the common factors in the numerator and denominator.
#26 [pic] (GCF=3) #27 [pic] (GCF=3)
Q: Simplify expression and state restrictions on variables.
#30 [pic].
#32 [pic], thus there are no restrictions on c.
Homework:
Pg. 542/ #6, 10, 14, 20, 21, 25, 31, 33, 35, 37, 39, 41, 43, 45
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