1. Chapter 7
Rational Expressions
7.1
7.2
7.3
7.4
7.5
7.6
7.7

2. Rational Functions and Multiplying and Dividing Rational Expressions
7.1
Rational Functions and Multiplying and Dividing
Rational Expressions
Objectives:
Find the domain of a rational function
Simplify rational expressions
Multiply and divide rational expressions

3. Examples of Rational Expressions
Rational expression is an expression that can be written as the quotient of two polynomials P and Q as long as Q is not 0.
Examples of Rational Expressions
A rational expression is undefined if the denominator is 0. If a variable in a rational expression is replaced with a number that makes the denominator 0, we say that the rational expression is undefined for this value of the variable.

4. Example 1 Find the domain of each rational expression. x – 2 ≠ 0 x ≠ 2
solve the equation “denominator ≠ 0”:
solve the equation “denominator ≠ 0”:
Since the denominator will never = 0
The domain is all real numbers
x – 2 ≠ 0
x ≠ 2

5. Example 2
Find the domain of the rational expression. Set the denominator ≠ 0. The domain of h is all real numbers except 2 and 3. x ≠ 2, x ≠ -3
x2 + x – 6 ≠ 0
(x – 2) (x + 3) ≠ 0
x – 2 ≠ or x + 3 ≠ 0
x ≠ or x ≠ -3

6. Simplifying or Writing a Rational Expression in Lowest Terms
1. Completely factor the numerator and denominator of the rational expression.
2. Divide out factors common to the numerator and denominator. (This is the same thing as “removing the factor of 1.”)

7. Example 3
Simplify each rational expression.

8. Example 4 Simplify each rational expression.
The terms in the numerator differ by the sign of the terms in the denominator. The polynomials are opposites of each other. Factor out a 1 from the numerator.
Simplify each rational expression.

9. Example 5
Multiply each.

10. Example 6
Perform the indicted operation.

11. Do the following now: 7.1 #5, 6, 10, 15, 18 7.1 Summary Objectives:
Find the domain of a rational function
Simplify rational expressions
Multiply and divide rational expressions
Do the following now:
#5, 6, 10, 15, 18

12. Adding and Subtracting Rational Expressions
7.2
Adding and Subtracting Rational Expressions
Objectives:
Add/subtract rational expressions with common denominators
Identify the least common denominator of two or more rational expressions
Add/subtract rational expressions with unlike denominators

13. Example 1
Add or subtract. a. b. c.

14. Example 2
Subtract.

15. To add or subtract rational expressions with unlike denominators, first write the rational expressions as equivalent rational expressions with common denominators.
The least common denominator (LCD) is usually the easiest common denominator to work with.
Finding the Least Common Denominator (LCD)
1. Factor each denominator completely,
2. The LCD is the product of all unique factors each raised to a power equal to the greatest number of times that the factor appears in any factored denominator.

16. Example 3 Find the LCD of the rational expressions.
Factor each denominator:
1st denom factored: 6y
2nd denom factored: 4(y + 3)
LCD: 12y(y + 3)

17. Example 4 Find the LCD of the rational expressions.
Factor each denominator:

18. Adding or Subtracting Rational Expressions with Unlike Denominators
1. Find the LCD of the rational expressions.
2. Write each rational expression as an equivalent rational expression whose denominator is the LCD found in Step 1.
3. Add or subtract numerators, and write the result over the common denominator.
4. Simplify resulting rational expression.

19. Example 5
Add. The LCD is 42a.
The LCD is (x + 3)(x – 3)

20. Example 6
Subtract.
–(2x – 6) 3 +

21. Example 7
Add

22. Do the following now: 7.2 #2, 4, 6, 8, 10 7.2 Summary Objectives:
Add/subtract rational expressions with common denominators
Identify the least common denominator of two or more rational expressions
Add/subtract rational expressions with unlike denominators
Do the following now:
#2, 4, 6, 8, 10

23. Simplifying Complex Fractions
7.3
Simplifying Complex Fractions
Objectives:
Simplify complex fractions by:
Simplifying the numerator and denominator and then dividing
Multiplying by a common denominator
Simplify expressions with negative exponents

24. A rational expression whose numerator, denominator, or both contain one or more rational expressions is called a complex rational expression or a complex fraction.

25. Example 1
Simplify the complex fraction. x

26. Example 2 Simplify: What is the LCD for all four fractions? LCD = x2y2 26

27. Example 3 Simplify the complex fraction.
What is the LCD for all four fractions?
LCD = 6y2

28. Example 4 Simplify. LCD = x2y2
When you have a rational expression where some of the variables have negative exponents, rewrite the expression using positive exponents.
Simplify.
What is the LCD for all three fractions?
LCD = x2y2

29. Do the following now: 7.3 #1, 3, 5, 7, 9,18 7.3 Summary Objectives:
Simplify complex fractions by:
Simplifying the numerator and denominator and then dividing
Multiplying by a common denominator
Simplify expressions with negative exponents
Do the following now:
#1, 3, 5, 7, 9,18

30. Dividing Polynomials: Long Division and Synthetic Division
7.4
Dividing Polynomials: Long Division and Synthetic Division
Objectives:
Divide a polynomial by a monomial
Divide polynomials using long division and synthetic division and be able to tell when to use which
Use the remainder theorem to evaluate polynomials
Use the factor theorem

31. Example 1
Divide

32. Dividing Polynomials
Dividing a polynomial by a polynomial other than a monomial uses a “long division” technique that is similar to the process known as long division in dividing two numbers, which is reviewed on the next slide.

33. Example 2 Divide 43 into 72. Multiply 1 times 43. Subtract 43 from 72.–
Bring down 5.
Divide 43 into 295. –
Multiply 6 times 43.
Subtract 258 from 295.
Bring down 6. –
Divide 43 into 376.
Multiply 8 times 43.
We then write our result as
Subtract 344 from 376.
Nothing to bring down.

34. As you can see from the previous example, there is a pattern in the long division technique.
Divide
Multiply
Subtract
Bring down
Then repeat these steps until you can’t bring down or divide any longer.
We will incorporate this same repeated technique with dividing polynomials.

35. 15 - 35 - x Example 3 So our answer is 4x – 5. Divide 7x into 28x2.
Multiply 4x times 7x+3.
Subtract 28x2 + 12x from 28x2 – 23x.
Bring down – 15. 15 -
Divide 7x into –35x. 35 - x
Multiply – 5 times 7x+3.
Subtract –35x–15 from –35x–15.
Nothing to bring down.
So our answer is 4x – 5.

36. Example 4 10 - 8 6 4 7 2 + - x x 14 4 + 8 + 20 - x 70 20 - x 78 10 - x
Divide 2x into 4x2. 8 6 4 7 2 + - x
Multiply 2x times 2x+7.
Subtract 4x2 + 14x from 4x2 – 6x. x 14 4 2 + 8 +
Bring down 8. 20 - x
Divide 2x into –20x. 70 20 - x
Multiply -10 times 2x+7.
Subtract –20x–70 from –20x+8. 78
Nothing to bring down. + 7 2 78 x 10 -
We write our final answer as

37. Example 5 Divide: – – 1. c2 by c. 2. Multiply c by c + 1.
3. Subtract c2 + c from c2 + 3c – 2. –
4. Bring down the next term –
5. Repeat the process until the degree of the remainder is less than the degree of the binomial divisor.
Remainder

38. Synthetic Division
To find the quotient and remainder when a polynomial of degree 1 or higher is divided by x – c, a shortened version of long division called synthetic division may be used.

39. Example 6
Use synthetic division to divide 2x3 – x2 – 13x + 1 by x – 3.
coefficients
Set divisor = 0 3 2 1
13 1 6 15 6
Always bring down the first # 2 5 2 7
remainder
Multiply
Then add
Then add …

40. Example 7
Use synthetic division to divide x4 – 3x3 – 13x2 + 6x + 32 by x + 3. 3 1
13 6 32 3 18
15 27 1 6 5 9 59

41. Add on to notes: Should you use long division or synthetic division?
1x /3 x x x 1
7/3 3 1
10/3
10/3
19/3 1
10/3
10/3
19/3
19/3

42. Use the remainder theorem and synthetic division to find P(3) if
If a polynomial P(x) is divided by x – c, then the remainder is P(c).
Example 8
Use the remainder theorem and synthetic division to find P(3) if c 3
18 32
15 9
27 15 45 90
270 3 9 5 15 30 90
270

43. Factor Theorem
A polynomial function f(x) has a factor of x – c, if and only if f(c) = 0.

44. Example 9 Given the polynomial equation 4 2 13 17 12 8 20 12 2 5
a. Use the Remainder Thm. to show that 4 is a solution of the equation.
b. Use the Factor Theorem to solve the polynomial equation.
proposed solution 4 2
13 17 12
Because the remainder is 0, then 4 is a solution of the given equation. 8
20
12 2 5 3
remainder

45. Do the following now: 7.4 #1, 3, 5, 7, 13,16 7.4 summary Objectives:
Divide a polynomial by a monomial
Divide polynomials using long division and synthetic division and be able to tell when to use which
Use the remainder theorem to evaluate polynomials
Use the factor theorem
Do the following now:
#1, 3, 5, 7, 13,16

46. Solving Equations Containing Rational Expressions
7.5
Solving Equations Containing Rational Expressions
Objectives:
Solve equations containing rational expressions.

47. First note that an equation contains an equal sign and an expression does not.
Equation Expression

48. Solving an Equation Containing Rational Expressions
1. Multiply both sides of the equation by the LCD of all rational expressions in the equation.
2. Simplify both sides.
3. Determine whether the equation is linear, quadratic, or higher degree and solve accordingly.
4. Check the solution in the original equation.

49. Example 1
LCD:
LCD is 6x
Restrictions:
6X ≠ 0
X ≠ 0
Solve:
Check: 2 x

50. Example 2 Solve: 3 2 LCD: LCD is 6X(X+1) Restrictions: 6X(X + 1) ≠ 03 2
Check restrictions

51. Example 3 Solve: LCD: LCD is 3(x+ 2)(x + 5) Restrictions:
Check restrictions

52. Example 4 Solve: Restrictions: (x – 1)(x + 1) ≠ 0 x ≠ 1, x ≠ -1
Check restrictions
Single fraction on each side I would cross multiply

53. Example 5 Solve: LCD: LCD is (3 – a)(3 + a) Restrictions:
Restriction: a ≠ 3 so the answer is no solution. 

54. Do the following now: 7.5 #3, 6, 9, 12, 15 7.5 summary Objectives:
Solve equations containing rational expressions
Find the LCD
Multiply both sides by the LCD
Solve the equation with skills from Algebra I
Don’t forget to check for the restrictions!!!!
Do the following now:
#3, 6, 9, 12, 15

55. Rational Equations and Problem Solving
7.6
Rational Equations and Problem Solving
Objectives:
Solve an equation containing rational expressions for a specified variable.
Solve problems by writing equations containing rational expressions.

56. Solving Equations for a Specified Variable
1. Clear the equation of fractions or rational expressions by multiplying each side of the equation by the least common denominator (LCD) of all denominators in the equation.
2. Use the distributive property to remove grouping symbols such as parentheses.
3. Combine like terms on each side of the equation.
4. Use the addition property of equality to rewrite the equation as an equivalent equation with terms containing the specified variable on one side and all other terms on the other side.
5. Use the distributive property and the multiplication property of equality to get the specified variable alone.

57. Example 1
LCD:
LCD is RR1R2
Solve: for R1.

58. Example 2
The quotient of a number and 9 times its reciprocal is 1. Find the number.
n = the number, then
= the reciprocal of the number.

59. The ratio of the numbers a and b can also be written as a:b, or .
Ratio is the quotient of two numbers or two quantities.
The ratio of the numbers a and b can also be written as a:b, or .
The units associated with the ratio are important.
The units should match.
If the units do not match, it is called a rate, rather than a ratio.
A proportion is a mathematical statement that two ratios are equal to each other.

60. Example 3 Solve the proportion for x. Restriction(s) x ≠ –2
Single fraction on each side of the equality sign, so cross multiple
Solve the proportion for x.
Restriction(s)
x ≠ –2
Check restrictions

61. Example 4
If a 170-pound person weighs approximately 65 pounds on Mars, how much does a 9000-pound satellite weigh on Mars?
person
satellite

62. Hours to Complete the Job Part of the Job Completed in 1 Hour
Example 5
An experienced roofer can roof a house in 26 hours. A beginner needs 39 hours to do the same job. How long will it take if the two roofers work together?
Let t = time in hours to complete the job together
Then 1/t = part of the job they each complete.
Hours to Complete the Job
Part of the Job Completed in 1 Hour
Beginning roofer 39
Experienced roofer 26
Together t

63. Example 6
The speed of Lazy River’s current is 5 mph. A boat travels 20 miles downstream in the same time as traveling 10 miles upstream. Find the speed of the boat in still water.
Rate
Time
Distance
Downstream
Upstream
r + 5 20
r – 5 10

64. Do the following now: 7.6 #4, 9, 10, 12, 13,17 7.6 summary Objectives:
Solve an equation containing rational expressions for a specified variable.
Solve problems by writing equations containing rational expressions.
Do the following now:
#4, 9, 10, 12, 13,17

65. Variation and Problem Solving
7.7
Variation and Problem Solving
Objectives:
Solve problems involving direct, inverse, joint, and combined variations.

66. Direct Variation
y varies directly as x, or y is directly proportional to x, if there is a nonzero constant k such that
y = kx
The number k is called the constant of variation or the constant of proportionality.

67. Example 1
If y varies directly as x, find the constant of variation k and the direct variation equation, given that y = 5 when x = 30.
y = kx
5 = k·30
k = 1/6
The constant of variation is 1/6.
The direct variation equation is
If y varies directly as x, and y = 48 when x = 6, then find y when x = 15.
y = kx
48 = k·6
8 = k
So the equation is y = 8x.
y = 8·15
y = 120

68. Example 2
At sea, the distance to the horizon is directly proportional to the square root of the elevation of the observer. If a person who is 36 feet above water can see 7.4 miles, find how far a person 64 feet above the water can see. Round your answer to two decimal places.
So our equation is

70. Example 3
If y varies inversely as x, find the constant of variation k and the inverse variation equation, given that y = 63 when x = 3. k = 63·3 k = 189 The inverse variation equation is

71. Joint Variation
If the ratio of a variable y to the product of two or more variables is constant, then y varies jointly as, or is jointly proportional to the other variables. If
y = kxz
then the number k is the constant or variation or the constant or proportionality.

72. Example 4
The maximum weight that a circular column can hold is inversely proportional to the square of its height. If an 8-foot column can hold 2 tons, find how much weight a 10-foot column can hold.
So our equation is

73. Do the following now: 7.7 #1, 4, 6, 10, 12,13 7.7 summary Directly
Objectives:
Solve problems involving direct, inverse, joint, and combined variations.
Directly
y = kx
Inversely
y = k/x
Jointly
y = kxz
Do the following now:
#1, 4, 6, 10, 12,13