1. Simplifying
Rational Expressions
Section 11.3

2. Simplifying Rational Expressions
The objective is to be able to simplify a rational expression

3. Vocabulary Polynomial – The sum or difference of monomials.
Rational expression – A fraction whose numerator and denominator are polynomials.
Reduced form – a rational expression in which the numerator and denominator have no factors in common.

4. Simplifying Rational Expressions
Divide out the common factors
Factor the numerator and denominator and then divide the common factors

5. Dividing Out Common Factors
Step 1 – Identify any factors which are common to both the numerator and the denominator.
The numerator and denominator have a common factor.
The common factor is the five.

6. Dividing Out Common Factors
Step 2 – Divide out the common factors.
The fives can be divided since 5/5 = 1
The x remains in the numerator.
The (x-7) remains in the denominator

7. Factoring the Numerator and Denominator
Factor the numerator.
Factor the denominator.
Divide out the common factors.
Write in simplified form.

8. Factoring
Step 1: Look for common factors to both terms in the numerator.
3 is a factor of both 3 and 9.
X is a factor of both x2 and x.
Step 2: Factor the numerator. 3 x ( x + 3 ) 3 12 x

9. Factoring
Step 3: Look for common factors to the terms in the denominator and factor.
The denominator only has one term The 12 and x3 can be factored.
The 12 can be factored into 3 x 4.
The x3 can be written as x • x2. 3 x ( x + 3 ) 2 3 4 x x

10. Divide and Simplify x + 3 4 x 2
Step 4: Divide out the common factors. In this case, the common factors divide to become 1.
Step 5: Write in simplified form. x + 3 2 4 x

11. You Try It
Simplify the following rational expressions.

12. Problem 1 Factor the numerator and denominator
Divide out the common factors.
Write in simplified form.

13. Problem 2 Factor the numerator and denominator
You Try It
Problem 2
Factor the numerator and denominator
Divide out the common factors.
Write in simplified form.

14. Problem 3 Factor the numerator and denominator
You Try It
Problem 3
Factor the numerator and denominator
Divide out the common factors.
Write in simplified form.

15. Problem 4 Factor the numerator and denominator
You Try It
Problem 4
Factor the numerator and denominator
Divide out the common factors.
Write in simplified form.

16. Problem 5 Factor the numerator and denominator
You Try It
Problem 5
Factor the numerator and denominator
Divide out the common factors.
Write in simplified form.

17. Factor

23. 12.1 – Simplifying Rational Expressions

24. 12.1 – Simplifying Rational Expressions

25. 12.1 – Simplifying Rational Expressions

26. Restrictions on Rational Expressions
For what value of x is undefined?
It is undefined for any value of “x” which makes the denominator zero.
The restriction is that x cannot equal 5.

27. YOU TRY IT
What are the excluded values of the variables for the following rational expressions?

28. Problem 1
Solution
y  0
z  0

29. Problem 2 Solution 2x - 12 = 0 ANSWER 2x - 12 + 12 = 0 + 12 x  6

30. Problem 3 Solution C2 + 2C - 8 = 0 Answer C  2 (C-2)(C+4) = 0 C  -4
C-2 = or C + 4 = 0
C-2+2 = C = 0 - 4
C = or C = -4
Answer
C  2
C  -4