1. Do Now
Factor completely. 1. 2. 3. 4.
18𝑥𝑦( 3𝑥 3 − 𝑦 3 )

2. Assignment: P.6 Rational Expressions
Day 2
8/23/16
P.6 Rational Expressions
Objectives: Simplify rational expressions and operations on rational expressions.
Assignment:

3. Rational Expressions Rational Expressions
A rational expression is the quotient of two polynomials.
The set of real numbers for which an algebraic expression is defined is the domain of the expression.
Because rational expressions indicate division and division by zero is undefined, we must exclude numbers from a rational expression’s domain that make the denominator zero.
DENOMINATOR can not = 0

4. Example: Excluding Numbers from the Domain
Rational Expressions
Find all the numbers that must be excluded from the domain of the rational expression:
To determine the numbers that must be excluded from the domain, examine the denominator.
We set each factor equal to zero to find the excluded values.

5. You DO
Find all the numbers that must be excluded from the domain.
1) 4 𝑥− 𝑥 𝑥 2 − 𝑥 𝑥 2 +11𝑥+10
(x-1)(x+1)
(x+10)(x+1)
1. x≠ 2
2. x ≠ -1, x ≠1
3. x ≠-10,
x ≠-1

6. Example: Simplifying Rational Expressions
Check for excluded values.
Because the denominator is

7. 𝑥 3 + 𝑥 2 𝑥+1 = 𝑥 2 , x≠-1 3. 𝑥 2 −1 𝑥 2 −2𝑥+1 2. 𝑥 2 +6𝑥+5 𝑥 2 −25
𝑥 3 + 𝑥 2 𝑥+1
= 𝑥 2 (𝑥+1) (𝑥+1)
You DO 1.
= 𝑥 2 , x≠-1
𝑥 2 +6𝑥+5 𝑥 2 −25
= (𝑥+5)(𝑥+1) (𝑥−5)(𝑥+5)
= 𝑥+1 𝑥−5 , x ≠ 5, x ≠ -5
3. 𝑥 2 −1 𝑥 2 −2𝑥+1
= (𝑥+1)(𝑥−1) (𝑥−1)(𝑥−1)
= 𝑥+1 𝑥−1 , 𝑥 ≠ 1

8. Multiplying Rational Expressions
1. Factor all numerators and denominators completely.
2. Divide numerators and denominators by common factors.
3. Multiply the remaining factors in the numerators and multiply the remaining factors in the denominators.

9. Example: Multiplying Rational Expressions
𝑥+3 𝑥 2 −4 ∗ 𝑥 2 −𝑥−6 𝑥 2 +6𝑥+9
Multiply:
Factor the numerator and the denominator:
Check for excluded values.
Because the denominator is ∗ ∗
Divide numerators and denominators by common factors: ∗ ∗

10. = (𝑥−3)(𝑥+3) 𝑥 2 ∗ 𝑥(𝑥−3) (𝑥+4)(𝑥−3) 𝑥 2 −9 𝑥 2 ∗ 𝑥 2 −3𝑥 𝑥 2 +𝑥−12
You Do
Factor everything
Exclude values of x
Cancel 1.
= (𝑥−3)(𝑥+3) 𝑥 2 ∗ 𝑥(𝑥−3) (𝑥+4)(𝑥−3)
𝑥 2 −9 𝑥 2 ∗ 𝑥 2 −3𝑥 𝑥 2 +𝑥−12 1
(𝑥−3)(𝑥+3) 𝑥(𝑥+4)
,X≠-4,0, 3

11. = ( 𝑥 2 +2𝑥+4) 3𝑥 𝑥 3 −8 𝑥 2 −4 ∗ 𝑥+2 3𝑥 𝑥≠−2, 0, 2 ,𝑥≠−2, 0, 2
𝑥 3 −8 𝑥 2 −4 ∗ 𝑥+2 3𝑥
= 𝑥 −2 ( 𝑥 2 +2𝑥+4) 𝑥−2 ( 𝑥+2) ∗ (𝑥+2) 3𝑥
𝑥≠−2, 0, 2
= 𝑥 −2 ( 𝑥 2 +2𝑥+4) 𝑥−2 ( 𝑥+2) ∗ (𝑥+2) 3𝑥
= ( 𝑥 2 +2𝑥+4) 3𝑥
,𝑥≠−2, 0, 2

12. Dividing Rational Expressions
The quotient of two rational expressions is the product of the first expression and the multiplicative inverse, or reciprocal, of the second expression.
The reciprocal is found by interchanging the numerator and the denominator.
Thus, we find the quotient of two rational expressions by inverting the divisor and multiplying.
Keep change Flip

13. Example: Dividing Rational Expressions
Divide:
Invert the divisor and multiply.
Factor as many numerators and denominators as possible. ● ● ●
Because the denominator is

14. Example: Dividing Rational Expressions (continued)
Divide:
Divide numerators and denominators by common factors.
= 3(𝑥−1 𝑥(𝑥+2 ,𝑥≠0,𝑥≠1,𝑥≠−2

15. 𝑥+2 𝑥−3 , 𝑥≠−3, 𝑥≠3, 𝑥≠4 1. 𝑥+3 𝑥 2 −4 ∗ 𝑥 2 −𝑥−6 𝑥 2 +6𝑥+9
Divide or Multiply
Change this problem
𝑥+3 𝑥 2 −4 ∗ 𝑥 2 −𝑥−6 𝑥 2 +6𝑥+9
𝑥 2 −2𝑥−8 𝑥 2 −9 ÷ 𝑥−4 𝑥+3
𝑥+2 𝑥−3 , 𝑥≠−3, 𝑥≠3, 𝑥≠4

16. 3. 𝑥 2 +𝑥−12 𝑥 2 +𝑥 −30 ∗ 𝑥 2 +5𝑥+6 𝑥 2 −2𝑥 −3 ÷ 𝑥+3 𝑥 2 +7𝑥+6
𝑥 2 +𝑥−12 𝑥 2 +𝑥 −30 ∗ 𝑥 2 +5𝑥+6 𝑥 2 −2𝑥 −3 ÷ 𝑥+3 𝑥 2 +7𝑥+6 4.
, x≠ -5,-2,-1, 1

17. 3𝑥 2 +4𝑥+1 3𝑥 2 −5𝑥−2 ÷ 𝑥 2 −2𝑥−3 −5𝑥 2 +25𝑥−30

18. Adding and Subtracting Rational Expressions with the Same Denominator
Adding and Subtracting Rational Expressions with Like Denominators
To add or subtract rational expressions with the same denominator:
1. Add or subtract the numerators.
2. Place this result over the common denominator.
3. Simplify, if possible.
Reminder: check for excluded values.

19. Example: Subtracting Rational Expressions with the Same Denominator
Adding and Subtracting Rational Expressions with Like Denominators
Subtract:
Check for excluded values:

20. Adding and Subtracting Rational Expressions with Unlike Denominators
Adding and Subtracting Rational Expressions That Have Different Denominators
Adding and Subtracting Rational Expressions with Unlike Denominators
1. Find the LCD of the rational expressions.
2. Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and the denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD.
3. Add or subtract numerators, placing the resulting expression over the LCD.
4. If possible, simplify the resulting rational expression.

21. Example: Subtracting Rational Expressions with Different Denominators
Adding and Subtracting Rational Expressions with Unlike Denominators
Subtract:
Step 1 Find the least common denominator.
The LCD is
Step 2 Write equivalent expressions with the LCD as denominators.

22. Example: Subtracting Rational Expressions with Different Denominators
Adding and Subtracting Rational Expressions with Unlike Denominators
Subtract:
Step 3 Subtract numerators, putting this difference over the LCD.
Step 4 If necessary, simplify.

23. You Do 1. 2.
𝑥−3 𝑥 2 −1 − 𝑥+2 𝑥−1

24. = 3𝑥 (𝑥+5)(𝑥−2) - 2𝑥 (𝑥+3)(𝑥−2) =
You Do
Find the LCD- exclude values for x
Make equivalent fractions using the LCD
Add or Subtract
Simplify
3𝑥 𝑥 2 +3𝑥−10 − 2𝑥 𝑥 2 +𝑥−6
= 3𝑥 (𝑥+5)(𝑥−2) 𝑥 (𝑥+3)(𝑥−2) =
3𝑥(𝑥+3) (𝑥+5)(𝑥−2)(𝑥+3)
− 2𝑥(𝑥+5) (𝑥+3)(𝑥−2)(𝑥+5) =
3𝑥 𝑥+3 −[2𝑥 𝑥+5 ] (𝑥+5)(𝑥−2)(𝑥+3)
= 3𝑥 2 +9𝑥− 2𝑥 2 −10𝑥 (𝑥+5)(𝑥−2)(𝑥+3) =
𝑥 2 −𝑥 (𝑥+5)(𝑥−2)(𝑥+3) 𝑥
≠ -5,-3,2

25. Do Now

26. Complex Rational Expressions
A complex fraction or complex rational expression is one that has AT LEAST one fraction in the NUMERATOR or Denominator or IN BOTH the numerator and denominator. Below are some examples.

27. There are two methods to simplify complex fractions
𝑥+ 1 𝑥 1 𝑥 + 3 𝑥 2

28. 𝑥 2 +1 𝑥 ÷ 𝑥+3 𝑥 2 = 𝑥+ 1 𝑥 1 𝑥 + 3 𝑥 2 = 𝑥 2 +1 𝑥 𝑥+3 𝑥 2
Method Simplify the numerator and the denominator
𝑥+ 1 𝑥 1 𝑥 + 3 𝑥 2 =
𝑥 2 +1 𝑥 𝑥+3 𝑥 2
Rewrite the problem
𝑥 2 +1 𝑥 ÷ 𝑥+3 𝑥 2 =

29. 𝑥 2 +1 𝑥 ÷ 𝑥+3 𝑥 2 = 𝑥 2 +1 𝑥 ∗ 𝑥 2 𝑥+3 = 𝑥 2 𝑥 2 +1 𝑥 𝑥+3
Rewrite the problem
𝑥 2 +1 𝑥 ÷ 𝑥+3 𝑥 2 =
𝑥 2 +1 𝑥 ∗ 𝑥 2 𝑥+3 = 𝑥 2 𝑥 𝑥 𝑥+3
= 𝑥 𝑥 (𝑥+3)

30. 𝑥+ 1 𝑥 1 𝑥 + 3 𝑥 2 𝑥+ 1 𝑥 ∗ 𝑥 2 ( 1 𝑥 + 3 𝑥 2 )∗ 𝑥 2 = 𝑥 3 +𝑥 𝑥+3
Method Find the LCD & MULTIPLY the numerator and denominator by the LCD
𝑥+ 1 𝑥 1 𝑥 + 3 𝑥 2
𝑥+ 1 𝑥 ∗ 𝑥 2 ( 1 𝑥 + 3 𝑥 2 )∗ 𝑥 2 =
𝑥 3 +𝑥 𝑥+3
= 𝑥 𝑥 𝑥+3

31. Example: Simplifying a Complex Rational Expression
Simplifying Rational Expressions
Simplify:

32. 2 3 1 2 1 𝑥 + 1 𝑦 𝑥+𝑦 Divide: Your Turn 1+ 1 𝑥 1− 1 𝑥
1+ 1 𝑥 1− 1 𝑥
= 𝑥 𝑥 + 1 𝑥 𝑥 𝑥 − 1 𝑥
= 𝑥+1 𝑥 𝑥−1 𝑥 1) 2)
1 𝑥 + 1 𝑦 𝑥+𝑦 3)

33. 4.

34. 5. 6.
3 𝑥−2 − 4 𝑥 𝑥 2 −4
1 𝑥 − 𝑥 + 3 4 7.
1 (𝑥+ℎ) 2 − 1 𝑥 2 ℎ

35. Fractional Expressions in Calculus
Fractional expressions containing radicals occur frequently in calculus. These expressions can often be simplified using the procedure for simplifying complex rational expressions
Simplify:

37. Your Turn:
9−𝑥 𝑥 −𝑥 −𝑥 2

38. Rationalizing the Numerator
To rationalize a numerator, multiply the numerator and the denominator by the conjugate of the numerator.
Rationalize the numerator:

39. 𝑥+ℎ − 𝑥 ℎ
h ≠0 but what happens to the expression as h approaches 0