1. Add, Subtract, multiply, and divide rational expressions
Lesson 4
Add, Subtract, multiply, and divide rational expressions

2. Warm up
Find the product.
(x+6)(x-6)
(7x-8)(7x+8)
(x+6)²

3. Warm up
Divide 5x³ + 20x² - 10x by 5x

4. What is a rational expression?
A rational expression is an expression that can be written as a ratio of two polynomials.
Ex. 8
-2x x
x2 – 3x
x + 5
x² + 2x - 15

5. Excluded values
A rational expression is undefined when the denominator is 0. A number that makes the denominator 0 is an excluded value.
Let’s look at the examples before…are there any values that should be excluded? 8
-2x x
x2 –3x
x + 2
x² + 2x - 15

6. Examples Find the excluded values, if any, of the expression. x -1
To find the excluded values, you have to factor the denominator (if possible). Then set each factor equal to ZERO and solve. 9x
5x - 15
x -1
x² - 16
x + 6
x² + 4x - 12

7. Simplest Form
A rational expression is in simplest form if the numerator and denominator have no factors in common other than 1.
Examples: Are these expressions in simplest form? 8
-2x x
3x - 9
x + 5
x² + 2x - 15

8. What about these? Simplify, if possible. State the excluded values.
7q² - 14 q
14q² 11
y + 6 2m
8m(m-1)

9. Try these… Simplify and state the excluded values. x² + 13x + 42

10. Guided Practice Pp. 437-438 1-3 (all parts)
Rational Expressions Handout

11. Multiplying Rational Expressions
½ * ¾ =
2/5 * 5/4 =
2/3 * 5/4 =
2x² * 1/x =
4xy/5 * 20y/4x²

12. Multiply and Divide Rational Expressions with variables
Factor the numerators and denominators.
Simplify where possible (…Cancel)
Multiply the numerators and denominators straight across.
List restrictions (excluded values) if any…
Example: x² + x – x² + 15x
10x² - 20x x² - 2x - 15 *

13. Examples * * 4x² (x+8) 2x³ + 10x – 48x x² - 1 4x – 2

14. Guided Practice
Textbook p

15. Let’s Divide
¾ divided by ½
2/5 divided by 2
1/3 divided by 4/5

16. So…to divide rational expressions…
Find the reciprocal of the second fraction.
Simplify according to your rules…
Multiply straight across…

17. Try these… ÷ 6 5x² 25x² 7x +21 21x + 63 30 20 ÷ x + 2 x² + 11x + 18
÷ 6
5x² x²
7x x + 63 ÷
x + 2 x² + 11x + 18
3x – 3 x - 1

18. Graphic Organizer
Fill in your graphic organizer “How do you multiply or divide rational expressions?”
Work each example in the space provided.

19. Practice Complete textbook pp. 441-444
Dividing rational expressions handout.
Multiplying rational exp handout.

20. Finding the LCM Find the LCM of 6x and 8x²
Step 1: Write the factors of each expression.
6x = 2 * 3 * x
8x² = 2 * 2 * 2 * x * x

21. Circle the factors that both expressions have in common…
Next…
Circle the factors that both expressions have in common…
Step 2:
6x = 2 * 3 * x
8x² = 2 * 2 *2 * x * x
Step 3: List the common factors once…and then list every other factor…
Step 4: MULTIPLY
2 * x * 3 * 2 * 2 * x =24 x²

22. Guided Practice
Least Common Multiple Practice Handout
Complete 1-3

23. Find the LCM of… x² + 2x – 8 and x² + 7x + 12
Step 1: List the factors of each polynomial
x² + 2x – 8 = (x-2)(x+4)
x² + 7x + 12 = (x+3)(x+4)
Step 2: Circle the common factors.
Step 3:List the common factor once and then the other factors…then multiply…
(x+4)(x-2)(x+3) There’s no need to multiply this out…Leave it as it is…

24. Guided Practice
Complete the remaining problems on the practice handout.

25. Add/Subtract Rationals
x x – 2
7x x
So, on these I need to add/subtract the numerators and just bring over the denominator, then simplify, if possible?
5x x – 9
3x – x – 4

26. What if the denominators are not the same?
12x² x5
Find the LCD of the denominators…aka LCM.
Multiply top and bottom by missing factors…
Now that the denominators are the same…follow your steps for adding fractions…

27. Try these…
18r² r³
2x x
3x³ x

28. What to do with these… 12 - 4x x+2 x-3 4 - 3 x² - 7x x 2x + x+ 4
x² - 7x x
2x x+ 4
x² - 3x x – 3

29. One more…
x² + 5x x² - 16

30. Fill in the graphic organizer for add/subt rational expressions.

31. Practice
Pp Student Text
More practice handouts 11.5 & 11.6