The document discusses mathematical expressions and polynomials. It provides examples of algebraic expressions and operations with polynomials, such as factoring polynomials. Factoring polynomials makes it easier to calculate outputs, simplify rational expressions, and solve equations. One example factors the polynomial 2x3 - 5x2 + 2x and shows it is easier to evaluate the factored form for different values of x than the original polynomial. The key purposes of factoring polynomials are to simplify calculations and operations.

1. Expressions
Math 260
Dr. Frank Ma
LA Harbor College

2. We order pizzas from Pizza Grande.
Expressions

3. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Expressions

4. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Hence if we ordered 5 pizzas delivered, the total cost
would be 8(5) + 10 = $50, excluding the tip.
Expressions

5. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Hence if we ordered 5 pizzas delivered, the total cost
would be 8(5) + 10 = $50, excluding the tip.
If we want x pizzas delivered, then the total cost is
given by the formula “8x + 10”.
Expressions

6. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Hence if we ordered 5 pizzas delivered, the total cost
would be 8(5) + 10 = $50, excluding the tip.
If we want x pizzas delivered, then the total cost is
given by the formula “8x + 10”.
Such a formula is called an expression.
Expressions

7. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Hence if we ordered 5 pizzas delivered, the total cost
would be 8(5) + 10 = $50, excluding the tip.
If we want x pizzas delivered, then the total cost is
given by the formula “8x + 10”.
Such a formula is called an expression.
Expressions
If we ordered x = 100 pizzas, the cost would be
8(100)+10 = $810.

8. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Hence if we ordered 5 pizzas delivered, the total cost
would be 8(5) + 10 = $50, excluding the tip.
If we want x pizzas delivered, then the total cost is
given by the formula “8x + 10”.
Such a formula is called an expression.
Expressions
If we ordered x = 100 pizzas, the cost would be
8(100)+10 = $810. The value x = 100 is called the
input and the projected cost $810 is called the output.

9. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Hence if we ordered 5 pizzas delivered, the total cost
would be 8(5) + 10 = $50, excluding the tip.
If we want x pizzas delivered, then the total cost is
given by the formula “8x + 10”.
Such a formula is called an expression.
Expressions
Definition: Mathematical expressions are calculation
procedures which are written with numbers, variables,
operation symbols +, –, *, / and ( )’s.
If we ordered x = 100 pizzas, the cost would be
8(100)+10 = $810. The value x = 100 is called the
input and the projected cost $810 is called the output.

10. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Hence if we ordered 5 pizzas delivered, the total cost
would be 8(5) + 10 = $50, excluding the tip.
If we want x pizzas delivered, then the total cost is
given by the formula “8x + 10”.
Such a formula is called an expression.
Expressions
Definition: Mathematical expressions are calculation
procedures which are written with numbers, variables,
operation symbols +, –, *, / and ( )’s.
Expressions calculate the expected future results.
If we ordered x = 100 pizzas, the cost would be
8(100)+10 = $810. The value x = 100 is called the
input and the projected cost $810 is called the output.

11. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions

12. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Trigonometric or log-formulas
are not algebraic.

13. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 – 2x + 4,

14. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 – 2x + 4,
x2 + 3
3 x3 – 2x – 4
,

15. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 – 2x + 4,
x2 + 3
3 x3 – 2x – 4
,
(x1/2 + y)1/3
(4y2 – (x + 4)1/2)1/4

16. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 – 2x + 4,
x2 + 3
3 x3 – 2x – 4
,
(x1/2 + y)1/3
(4y2 – (x + 4)1/2)1/4
Examples of non–algebraic expressions are
sin(x), 2x, log(x + 1).

17. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 – 2x + 4,
x2 + 3
3 x3 – 2x – 4
,
(x1/2 + y)1/3
(4y2 – (x + 4)1/2)1/4
Examples of non–algebraic expressions are
sin(x), 2x, log(x + 1).
The algebraic expressions anxn + an–1xn–1...+ a1x + a0
where ai are numbers, are called polynomials (in x).

18. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 – 2x + 4,
x2 + 3
3 x3 – 2x – 4
,
(x1/2 + y)1/3
(4y2 – (x + 4)1/2)1/4
Examples of non–algebraic expressions are
sin(x), 2x, log(x + 1).
The algebraic expressions anxn + an–1xn–1...+ a1x + a0
where ai are numbers, are called polynomials (in x).
The algebraic expressions where P and Q are
polynomials, are called rational expressions.
P
Q

19. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.

20. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x – 5)(x +3) – (3x – 4)(x + 5)

21. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x – 5)(x +3) – (3x – 4)(x + 5)
The point of this problem is
how to subtract a “product”.

22. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert [ ]
The point of this problem is
how to subtract a “product”.

23. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x – 5)(x +3) – [(3x – 4)(x + 5)]
= 2x2 + x – 15 – [3x2 + 11x – 20]
Insert [ ]
remove [ ]

24. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x – 5)(x +3) – [(3x – 4)(x + 5)]
= 2x2 + x – 15 – [3x2 + 11x – 20]
= 2x2 + x – 15 – 3x2 – 11x + 20
Insert [ ]
remove [ ]

25. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x – 5)(x +3) – [(3x – 4)(x + 5)]
= 2x2 + x – 15 – [3x2 + 11x – 20]
= 2x2 + x – 15 – 3x2 – 11x + 20
= –x2 – 10x + 5
Insert [ ]
remove [ ]

26. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x – 5)(x +3) – [(3x – 4)(x + 5)]
= 2x2 + x – 15 – [3x2 + 11x – 20]
= 2x2 + x – 15 – 3x2 – 11x + 20
= –x2 – 10x + 5
Insert [ ]
remove [ ]

27. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x – 5)(x +3) – [(3x – 4)(x + 5)]
= 2x2 + x – 15 – [3x2 + 11x – 20]
= 2x2 + x – 15 – 3x2 – 11x + 20
= –x2 – 10x + 5
Insert [ ]
remove [ ]
(2x – 5)(x +3) – (3x – 4)(x + 5)
= (2x – 5)(x +3) + (–3x + 4)(x + 5)
= …
Or distribute the minus sign and
change it to an addition problem:

28. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x – 5)(x +3) – [(3x – 4)(x + 5)]
= 2x2 + x – 15 – [3x2 + 11x – 20]
= 2x2 + x – 15 – 3x2 – 11x + 20
= –x2 – 10x + 5
Insert [ ]
To factor an expression
means to write it as a product
in a non-obvious way.
remove [ ]
(2x – 5)(x +3) – (3x – 4)(x + 5)
= (2x – 5)(x +3) + (–3x + 4)(x + 5)
= …
Or distribute the minus sign and
change it to an addition problem:

29. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x – 5)(x +3) – [(3x – 4)(x + 5)]
= 2x2 + x – 15 – [3x2 + 11x – 20]
= 2x2 + x – 15 – 3x2 – 11x + 20
= –x2 – 10x + 5
Insert [ ]
A3 B3 = (A B)(A2 AB + B2)
Important Factoring Formulas:
To factor an expression
means to write it as a product
in a non-obvious way.
A2 – B2 = (A + B)(A – B)
+–
+– +–
remove [ ]
(2x – 5)(x +3) – (3x – 4)(x + 5)
= (2x – 5)(x +3) + (–3x + 4)(x + 5)
= …
Or distribute the minus sign and
change it to an addition problem:

30. Example B. Factor 64x3 + 125
Polynomial Expressions
A3 B3 = (A B)(A2 AB + B2)+– +
–+–

31. Example B. Factor 64x3 + 125
64x3 + 125
= (4x)3 + (5)3
Polynomial Expressions
A3 B3 = (A B)(A2 AB + B2)+– +
–+–
A3 B3

32. Example B. Factor 64x3 + 125
64x3 + 125
= (4x)3 + (5)3
= (4x + 5)((4x)2 – (4x)(5) +(5)2)
Polynomial Expressions
A3 B3 = (A B)(A2 AB + B2)+– +
–+–
(A B) (A2 AB + B2)+ –

33. Example B. Factor 64x3 + 125
64x3 + 125
= (4x)3 + (5)3
= (4x + 5)((4x)2 – (4x)(5) +(5)2)
= (4x + 5)(16x2 – 20x + 25)
Polynomial Expressions
A3 B3 = (A B)(A2 AB + B2)+– +
–+–

34. Example B. Factor 64x3 + 125
64x3 + 125
= (4x)3 + (5)3
= (4x + 5)((4x)2 – (4x)(5) +(5)2)
= (4x + 5)(16x2 – 20x + 25)
Polynomial Expressions
We factor polynomials for the following purposes.
A3 B3 = (A B)(A2 AB + B2)+– +
–+–

35. Example B. Factor 64x3 + 125
64x3 + 125
= (4x)3 + (5)3
= (4x + 5)((4x)2 – (4x)(5) +(5)2)
= (4x + 5)(16x2 – 20x + 25)
Polynomial Expressions
We factor polynomials for the following purposes.
I. It’s easier to calculate an output or to check
the sign of an output using the factored form.
A3 B3 = (A B)(A2 AB + B2)+– +
–+–

36. Example B. Factor 64x3 + 125
64x3 + 125
= (4x)3 + (5)3
= (4x + 5)((4x)2 – (4x)(5) +(5)2)
= (4x + 5)(16x2 – 20x + 25)
Polynomial Expressions
We factor polynomials for the following purposes.
I. It’s easier to calculate an output or to check
the sign of an output using the factored form.
II. To simplify or perform algebraic operations with
rational expressions.
A3 B3 = (A B)(A2 AB + B2)+– +
–+–

37. Example B. Factor 64x3 + 125
64x3 + 125
= (4x)3 + (5)3
= (4x + 5)((4x)2 – (4x)(5) +(5)2)
= (4x + 5)(16x2 – 20x + 25)
Polynomial Expressions
We factor polynomials for the following purposes.
I. It’s easier to calculate an output or to check
the sign of an output using the factored form.
II. To simplify or perform algebraic operations with
rational expressions.
III. To solve equations (See next section).
A3 B3 = (A B)(A2 AB + B2)+– +
–+–

38. Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions.
Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3
by factoring it first.

39. Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.
Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3
by factoring it first.

40. Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3
by factoring it first.
Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.

41. Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3
by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.

42. Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3
by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
Plug in x = –2:
–2 [2(–2) – 1] [(–2) – 2]
Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.

43. Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3
by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
Plug in x = –2:
–2 [2(–2) – 1] [(–2) – 2] = –2 [–5] [–4] = –40
Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.

44. Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3
by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
Plug in x = –2:
–2 [2(–2) – 1] [(–2) – 2] = –2 [–5] [–4] = –40
Plug in x = –1:
–1 [2(–1) – 1] [(–1) – 2]
Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.

45. Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3
by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
Plug in x = –2:
–2 [2(–2) – 1] [(–2) – 2] = –2 [–5] [–4] = –40
Plug in x = –1:
–1 [2(–1) – 1] [(–1) – 2] = –1 [–3] [–3] = –9
Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.

46. Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3
by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
Plug in x = –2:
–2 [2(–2) – 1] [(–2) – 2] = –2 [–5] [–4] = –40
Plug in x = –1:
–1 [2(–1) – 1] [(–1) – 2] = –1 [–3] [–3] = –9
Plug in x = 3:
3 [2(3) – 1] [(3) – 2] = 3 [5] [1] = 15
Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.

47. Determine the Signs of the Outputs.
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.

48. Example D. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2.
Determine the Signs of the Outputs.
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.

49. Example D. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2.
x2 – 2x – 3 = (x – 3)(x + 1).
Determine the Signs of the Outputs.
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.

50. Example D. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2.
x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2,
we get (–3/2 – 3)(–3/2 + 1)
Determine the Signs of the Outputs.
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.

51. Example D. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2.
x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2,
we get (–3/2 – 3)(–3/2 + 1) is (–)(–) = + .
Determine the Signs of the Outputs.
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.

52. Example D. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2.
x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2,
we get (–3/2 – 3)(–3/2 + 1) is (–)(–) = + .
Determine the Signs of the Outputs.
Rational Expressions
We say a rational expression is in the factored form
if it's numerator and denominator are factored.
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.

53. Example D. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2.
x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2,
we get (–3/2 – 3)(–3/2 + 1) is (–)(–) = + .
Determine the Signs of the Outputs.
Rational Expressions
We say a rational expression is in the factored form
if it's numerator and denominator are factored.
Example E. Factor
x2 – 1
x2 – 3x+ 2
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.

54. Example D. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2.
x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2,
we get (–3/2 – 3)(–3/2 + 1) is (–)(–) = + .
Determine the Signs of the Outputs.
Rational Expressions
We say a rational expression is in the factored form
if it's numerator and denominator are factored.
Example E. Factor
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
is the factored form.
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.

55. Rational Expressions
We put rational expressions in the factored form
in order to reduce, multiply or divide them.

56. Rational Expressions
We put rational expressions in the factored form
in order to reduce, multiply or divide them.
Cancellation Rule: Given a rational expression in
the factored form, common factors may be cancelled,
i.e.
x*y
x*z =
x*y
x*z =
y
z
1

57. Rational Expressions
We put rational expressions in the factored form
in order to reduce, multiply or divide them.
x*y
x*z =
x*y
x*z =
y
z
A rational expression that can't be cancelled any
further is said to be reduced.
Cancellation Rule: Given a rational expression in
the factored form, common factors may be cancelled,
i.e.

58. Rational Expressions
We put rational expressions in the factored form
in order to reduce, multiply or divide them.
Example F. Reduce x2 – 1
x2 – 3x+ 2
x*y
x*z =
x*y
x*z =
y
z
A rational expression that can't be cancelled any
further is said to be reduced.
Cancellation Rule: Given a rational expression in
the factored form, common factors may be cancelled,
i.e.

59. Rational Expressions
We put rational expressions in the factored form
in order to reduce, multiply or divide them.
Example F. Reduce x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2 =
(x – 1)(x + 1)
(x – 1)(x – 2)
x*y
x*z =
x*y
x*z =
y
z
A rational expression that can't be cancelled any
further is said to be reduced.
factor
Cancellation Rule: Given a rational expression in
the factored form, common factors may be cancelled,
i.e.

60. Rational Expressions
We put rational expressions in the factored form
in order to reduce, multiply or divide them.
Example F. Reduce x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2 =
(x – 1)(x + 1)
(x – 1)(x – 2)
x*y
x*z =
x*y
x*z =
y
z
A rational expression that can't be cancelled any
further is said to be reduced.
= (x + 1)
(x – 2)
factor
Cancellation Rule: Given a rational expression in
the factored form, common factors may be cancelled,
i.e.

61. Rational Expressions
Multiplication Rule:

62. Rational Expressions
Multiplication Rule:
P
Q
R
S
* = P*R
Q*S

63. Rational Expressions
Multiplication Rule:
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
÷

64. Rational Expressions
Multiplication Rule:
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
÷ = P*S
Q*R
Reciprocate

65. Rational Expressions
Multiplication Rule:
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
÷ = P*S
Q*R
Reciprocate
Example G. Simplify (2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)

66. Rational Expressions
Multiplication Rule:
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
÷ = P*S
Q*R
Reciprocate
Example G. Simplify (2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)
(2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)
=
(2x – 6)
(y + 3)
(y2 + 2y – 3)
(9 – x2)*
Reciprocate

67. Rational Expressions
Multiplication Rule:
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
÷ = P*S
Q*R
Reciprocate
Example G. Simplify (2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)
(2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)
=
(2x – 6)
(y + 3)
(y2 + 2y – 3)
(9 – x2)*
Reciprocate
To carry out these operations, put the expressions
in factored form and cancel as much as possible.

68. Rational Expressions
Multiplication Rule:
To carry out these operations, put the expressions
in factored form and cancel as much as possible.
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
÷ = P*S
Q*R
Reciprocate
Example G. Simplify (2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)
(2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)
=
(2x – 6)
(y + 3)
(y2 + 2y – 3)
(9 – x2)*
=
2(x – 3)
(y + 3)
(y + 3)(y – 1)
(3 – x)(3 + x)* factor and cancel

69. Rational Expressions
Multiplication Rule:
To carry out these operations, put the expressions
in factored form and cancel as much as possible.
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
÷ = P*S
Q*R
Reciprocate
Example G. Simplify (2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)
(2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)
=
(2x – 6)
(y + 3)
(y2 + 2y – 3)
(9 – x2)*
=
2(x – 3)
(y + 3)
(y + 3)(y – 1)
(3 – x)(3 + x)*
1
factor and cancel

70. Rational Expressions
Multiplication Rule:
To carry out these operations, put the expressions
in factored form and cancel as much as possible.
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
÷ = P*S
Q*R
Reciprocate
Example G. Simplify (2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)
(2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)
=
(2x – 6)
(y + 3)
(y2 + 2y – 3)
(9 – x2)*
=
2(x – 3)
(y + 3)
(y + 3)(y – 1)
(3 – x)(3 + x)*
–1 1
factor and cancel

71. Rational Expressions
Multiplication Rule:
To carry out these operations, put the expressions
in factored form and cancel as much as possible.
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
÷ = P*S
Q*R
Reciprocate
Example G. Simplify (2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)
(2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)
=
(2x – 6)
(y + 3)
(y2 + 2y – 3)
(9 – x2)*
=
2(x – 3)
(y + 3)
(y + 3)(y – 1)
(3 – x)(3 + x)*
–1 1
=
–2(y – 1)
(x + 3)

72. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions

73. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
II. to simplify complex fractions

74. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
II. to simplify complex fractions
III. to solve rational equations (later)

75. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ –
16
9
II. to simplify complex fractions
III. to solve rational equations (later)

76. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ –
16
9
Combining Rational Expressions (LCD Method):
II. to simplify complex fractions
III. to solve rational equations (later)

77. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ –
16
9
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ± G),
multiple (F ± G)* LCD/LCD,
II. to simplify complex fractions
III. to solve rational equations (later)

78. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ –
16
9
The LCD = 48,
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ± G),
multiple (F ± G)* LCD/LCD,
II. to simplify complex fractions
III. to solve rational equations (later)

79. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ –
16
9
The LCD = 48, ( )*
48
7
12
5
8
+ –
16
9
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ± G),
multiple (F ± G)* LCD/LCD,
48
II. to simplify complex fractions
III. to solve rational equations (later)

80. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ –
16
9
The LCD = 48, ( )*
48
7
12
5
8
+ –
16
9 48
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ± G),
multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD
and simplify (F ± G)(LCD) / LCD.
II. to simplify complex fractions
III. to solve rational equations (later)

81. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ –
16
9
The LCD = 48, ( )*
48
7
12
5
8
+ –
16
94
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ± G),
multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD
and simplify (F ± G)(LCD) / LCD.
48
II. to simplify complex fractions
III. to solve rational equations (later)

82. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ –
16
9
The LCD = 48, ( )*
48
67
12
5
8
+ –
16
94
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ± G),
multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD
and simplify (F ± G)(LCD) / LCD.
48
II. to simplify complex fractions
III. to solve rational equations (later)

83. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ –
16
9
The LCD = 48, ( )*
48
67
12
5
8
+ –
16
94 3
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ± G),
multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD
and simplify (F ± G)(LCD) / LCD.
48
II. to simplify complex fractions
III. to solve rational equations (later)

84. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ –
16
9
The LCD = 48, ( )*
48
67
12
5
8
+ –
16
94 3
=
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ± G),
multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD
and simplify (F ± G)(LCD) / LCD.
48 28 + 30 – 27
48
II. to simplify complex fractions
III. to solve rational equations (later)

85. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ –
16
9
The LCD = 48, ( )*
48
67
12
5
8
+ –
16
94 3
=
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ± G),
multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD
and simplify (F ± G)(LCD) / LCD.
48 28 + 30 – 27
48
=
48
31
II. to simplify complex fractions
III. to solve rational equations (later)

86. Rational Expressions
–
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3Example I. Combine

87. Rational Expressions
–
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Example I. Combine

88. Rational Expressions
–
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3),
Example I. Combine

89. Rational Expressions
–
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3),
multiplying LCD/LCD (= 1) to the problem,
Example I. Combine

90. Rational Expressions
–
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3),
multiplying LCD/LCD (= 1) to the problem,
–
(y2 + 2y – 3)(y – 1)(y + 2)
2y – 1 y – 3[ ]
Example I. Combine

91. Rational Expressions
–
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3),
multiplying LCD/LCD (= 1) to the problem,
–
(y2 + 2y – 3)(y – 1)(y + 2)
2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3)
Example I. Combine
LCD

92. Rational Expressions
–
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3),
multiplying LCD/LCD (= 1) to the problem,
–
(y – 1)(y + 2)
2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3)
Example I. Combine
LCD
expand and simplify.
(y – 1)(y + 3)

93. Rational Expressions
–
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3),
multiplying LCD/LCD (= 1) to the problem,
–
(y – 1)(y + 2)
2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3)
(y + 3)
Example I. Combine
LCD
expand and simplify.
(y – 1)(y + 3)

94. Rational Expressions
–
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3),
multiplying LCD/LCD (= 1) to the problem,
–
(y – 1)(y + 2)
2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3)
(y + 3) (y + 2)
Example I. Combine
LCD
expand and simplify.
(y – 1)(y + 3)

95. Rational Expressions
–
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3),
multiplying LCD/LCD (= 1) to the problem,
–
(y – 1)(y + 2)
2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3)
= (2y – 1)(y + 3) – (y – 3)(y + 2)
(y + 3) (y + 2)
Example I. Combine
LCD
LCD
expand and simplify.
(y – 1)(y + 3)

96. Rational Expressions
–
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3),
multiplying LCD/LCD (= 1) to the problem,
–
(y – 1)(y + 2)
2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3)
= (2y – 1)(y + 3) – (y – 3)(y + 2) = y2 + 6y + 3
(y + 3) (y + 2)
Example I. Combine
LCD
LCDLCD
expand and simplify.
(y – 1)(y + 3)

97. Rational Expressions
–
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2)
y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3),
multiplying LCD/LCD (= 1) to the problem,
–
(y – 1)(y + 2)
2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3)
= (2y – 1)(y + 3) – (y – 3)(y + 2) = y2 + 6y + 3
So –
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3
=
y2 + 6y + 3
(y – 1)(y + 2)(y + 3)
(y + 3) (y + 2)
Example I. Combine
LCD
LCDLCD
expand and simplify.
(y – 1)(y + 3)

98. Rational Expressions
Example J. Simplify
–3
1
A complex fraction is a fraction of fractions.
2
3
–4
1
3
2

99. Rational Expressions
Example J. Simplify
–3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
–4
1
3
2

100. Rational Expressions
Example J. Simplify
–3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
–4
1
3
2
3
1
2
3
4
1
3
2
.,,, Their LCD is 12.

101. Rational Expressions
Example J. Simplify
–3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
–4
1
3
2
3
1
2
3
4
1
3
2
.,,,
Multiplying 12/12 (or 1) to the problem:
–3
1
2
3
–4
1
3
2
Their LCD is 12.

102. Rational Expressions
Example J. Simplify
–3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
–4
1
3
2
3
1
2
3
4
1
3
2
.,,,
Multiplying 12/12 (or 1) to the problem:
–3
1
2
3
–4
1
3
2
( )
)(
12
12
Their LCD is 12.
=
1

103. Rational Expressions
Example J. Simplify
–3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
–4
1
3
2
3
1
2
3
4
1
3
2
.,,,
Multiplying 12/12 (or 1) to the problem:
–3
1
2
3
–4
1
3
2
( )
)(
12
12
=
Their LCD is 12.
–3
1
2
3
–4
1
3
2
*12
*12
*12
*12

104. Rational Expressions
Example J. Simplify
–3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
–4
1
3
2
3
1
2
3
4
1
3
2
.,,,
Multiplying 12/12 (or 1) to the problem:
–3
1
2
3
–4
1
3
2
( )
)(
12
12
=
Their LCD is 12.
–3
1
2
3
–4
1
3
2
*12
*12 *12
*12 4

105. Rational Expressions
Example J. Simplify
–3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
–4
1
3
2
3
1
2
3
4
1
3
2
.,,,
Multiplying 12/12 (or 1) to the problem:
–3
1
2
3
–4
1
3
2
( )
)(
12
12
=
Their LCD is 12.
–3
1
2
3
–4
1
3
2
*12
*12 *12
*12 4 6

106. Rational Expressions
Example J. Simplify
–3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
–4
1
3
2
3
1
2
3
4
1
3
2
.,,,
Multiplying 12/12 (or 1) to the problem:
–3
1
2
3
–4
1
3
2
( )
)(
12
12
=
Their LCD is 12.
–3
1
2
3
–4
1
3
2
*12
*12 *12
*12 4 6
3 4

107. Rational Expressions
Example J. Simplify
–3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
–4
1
3
2
3
1
2
3
4
1
3
2
.,,,
Multiplying 12/12 (or 1) to the problem:
–3
1
2
3
–4
1
3
2
( )
)(
12
12
=
Their LCD is 12.
–3
1
2
3
–4
1
3
2
*12
*12 *12
*12 4 6
3 4
= 3
4 – 18
– 8

108. Rational Expressions
Example J. Simplify
–3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
–4
1
3
2
3
1
2
3
4
1
3
2
.,,,
Multiplying 12/12 (or 1) to the problem:
–3
1
2
3
–4
1
3
2
( )
)(
12
12
=
Their LCD is 12.
–3
1
2
3
–4
1
3
2
*12
*12 *12
*12 4 6
3 4
= 3
4 – 18
– 8 = 14
5

109. Rational Expressions
Example K. Simplify
–
(x – h)
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
(x + h)
1
2h
Multiply the top and bottom by (x – h)(x + h) to reduce the
expression in the numerators to polynomials.
–
(x – h)
1
(x + h)
1
2h
=
–
(x – h)
1
(x + h)
1
2h
(x + h)(x – h)[ ]
(x + h)(x – h)*
=
–(x + h) (x – h)
2h(x + h)(x – h)
=
2h
2h(x + h)(x – h)
=
1
(x + h)(x – h)

110. To rationalize radicals in expressions we often use
the formula (x – y)(x + y) = x2 – y2.
Rationalize Radicals

111. To rationalize radicals in expressions we often use
the formula (x – y)(x + y) = x2 – y2.
(x + y) and (x – y) are called conjugates.
Rationalize Radicals

112. Example K: Rationalize the numerator
To rationalize radicals in expressions we often use
the formula (x – y)(x + y) = x2 – y2.
(x + y) and (x – y) are called conjugates.
h
x + h – x
Rationalize Radicals

113. To rationalize radicals in expressions we often use
the formula (x – y)(x + y) = x2 – y2.
(x + y) and (x – y) are called conjugates.
Rationalize Radicals
h
x + h – x
= h
(x + h – x) (x + h + x)
(x + h + x)
*
Example K: Rationalize the numerator h
x + h – x

114. To rationalize radicals in expressions we often use
the formula (x – y)(x + y) = x2 – y2.
(x + y) and (x – y) are called conjugates.
Rationalize Radicals
h
x + h – x
= h
(x + h – x) (x + h + x)
(x + h + x)
*
=
h
(x + h)2 – (x)2
(x + h + x)
Example K: Rationalize the numerator h
x + h – x

115. To rationalize radicals in expressions we often use
the formula (x – y)(x + y) = x2 – y2.
(x + y) and (x – y) are called conjugates.
Rationalize Radicals
h
x + h – x
= h
(x + h – x) (x + h + x)
(x + h + x)
*
=
h
(x + h)2 – (x)2
(x + h + x)
Example K: Rationalize the numerator h
x + h – x
(x + h) – (x) = h

116. To rationalize radicals in expressions we often use
the formula (x – y)(x + y) = x2 – y2.
(x + y) and (x – y) are called conjugates.
Rationalize Radicals
h
x + h – x
= h
(x + h – x) (x + h + x)
(x + h + x)
*
=
h
(x + h)2 – (x)2
(x + h + x)
=
h
h
(x + h + x)
Example K: Rationalize the numerator h
x + h – x
(x + h) – (x) = h

117. To rationalize radicals in expressions we often use
the formula (x – y)(x + y) = x2 – y2.
(x + y) and (x – y) are called conjugates.
Rationalize Radicals
h
x + h – x
= h
(x + h – x) (x + h + x)
(x + h + x)
*
=
h
(x + h)2 – (x)2
(x + h + x)
=
h
h
(x + h + x)
=
1
x + h + x
Example K: Rationalize the numerator h
x + h – x
(x + h) – (x) = h

118. Exercise A. Factor each expression then use the factored
form to evaluate the given input values. No calculator.
Applications of Factoring
1. x2 – 3x – 4, x = –2, 3, 5 2. x2 – 2x – 15, x = –1, 4, 7
3. x2 – x – 2, x = ½ ,–2, –½ 4. x3 – 2x2, x = –2, 2, 4
5. x4 – 3x2, x = –1, 1, 5 6. x3 – 4x2 – 5x, x = –4, 2, 6
B. Determine if the output is positive or negative using the
factored form.
7.
x2 – 4
x + 4
8. x3 – 2x2
x2 – 2x + 1
, x = –3, 1, 5 , x = –0.1, 1/2, 5
4.
x2 – 4
x + 4 5. x2 + 2x – 3
x2 + x
6. x3 – 2x2
x2 – 2x + 1
, x = –3.1, 1.9 , x = –0.1, 0.9, 1.05
, x = –0.1, 0.99, 1.01
1. x2 – 3x – 4, x = –2½, –2/3, 2½, 5¼
2. –x2 + 2x + 8, x = –2½, –2/3, 2½, 5¼
3. x3 – 2x2 – 8x, x = –4½, –3/4, ¼, 6¼,

119. C. Simplify. Do not expand the results.
Multiplication and Division of Rational Expressions
1. 10x *
2
5x3
15x
4
*
16
25x42. 3.12x6 *
5
6x14
4. 75x
49
*
42
25x3
5. 2x – 4
2x + 4
5x + 10
3x – 6
6.
x + 4
–x – 4
4 – x
x – 4
7. 3x – 9
15x – 5
3 – x
5 – 15x
8. 42 – 6x
–2x + 14
4 – 2x
–7x + 14
*
*
*
*
9.
(x2 – x – 2 )
(x2 – 1) (x2 + 2x + 1)
(x2 + x )
* 10.
(x2 + 5x – 6 )
(x2 + 5x + 6) (x2 – 5x – 6 )
(x2 – 5x + 6)
*
11. (x2 – 3x – 4 )
(x2 – 1) (x2 – 2x – 8)
(x2 – 3x + 2)
*
12. (– x2 + 6 – x )
(x2 + 5x + 6) (x2 – x – 12)
(6 – x2 – x)
*
13.(3x2 – x – 2)
(x2 – x + 2) (3x2 + 4x + 1)
(–x – 3x2)
14. (x + 1 – 6x2)
(–x2 – 4)
(2x2 + x – 1 )
(x2 – 5x – 6)
15. (x3 – 4x)
(–x2 + 4x – 4)
(x2 + 2)
(–x + 2)
16. (–x3 + 9x ) (x2 + 6x + 9)
(x2 + 3x) (–3x2 – 9x)
÷
÷
÷
÷

120. Multiplication and Division of Rational Expressions
D. Multiply, expand and simplify the results.
1. x + 3
x + 1
(x2 – 1) 2. x – 3
x – 2
(x2 – 4) 3. 2x + 3
1 – x
(x2 – 1)
4. 3 – 2x
x + 2
(x + 2)(x +1) 5.
3 – 2x
2x – 1
(3x + 2)(1 – 2x)
6. x – 2
x – 3
( x + 1
x + 3)( x – 3)(x + 3)
7. 2x – 1
x + 2
( – x + 2
2x – 3 )( 2x – 3)(x + 2)
+
8.
x – 2
x – 3
( x + 1
x + 3
) ( x – 3)(x + 3)–
9.
x – 2
x2 – 9
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 3)(x + 1)
10.
x + 3
x2 – 4
( – 2x + 1
x2 + x – 2
) ( x – 2)(x + 2)(x – 1)
11.
x – 1
x2 – x – 6
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 2)(x + 1)

121. E. Combine and simplify the answers.
–3
x – 3
+ 2x
–6 – 2x
3. 2x – 3
x – 3
– 5x + 4
5 – 15x
4.
3x + 1
6x – 4
– 2x + 3
2 – 3x5.
–5x + 7
3x – 12+
4x – 3
–2x + 86.
3x + 1
+
x + 3
4 – x211. x2 – 4x + 4
x – 4
+
x + 5
–x2 + x + 2
12.
x2 – x – 6
3x + 1
+
2x + 3
9 – x213.
x2 – x – 6
3x – 4
–
2x + 5
x2 – x – 6
14.
–x2 + 5x + 6
3x + 4
+
2x – 3
–x2 – 2x + 3
15.
x2 – x
5x – 4
–
3x – 5
1 – x216.
x2 + 2x – 3
–3
2x – 1
+ 2x
2 – 4x
1.
2x – 3
x – 2
+
3x + 4
5 – 10x
2.
3x + 1
2x – 5
– 2x + 3
5 – 10x
9.
–3x + 2
3x – 12
+
7x – 2
–2x + 8
10.
3x + 5
3x –2
– x + 3
2 – 3x7. –5x + 7
3x – 4 + 4x – 3
–6x + 88.
Addition and Subtraction of Fractions

122. Complex Fractions
1
2x + 1
– 2
3 –
1
2x + 1
3.
–2
2x + 1
–
+
3
x + 4
4.
1
x + 4
2
2x + 1
4
2x + 3
–
+
3
x + 4
5.
3
3x – 2
5
3x – 2
–5
2x + 5
–
+ 3
–x + 4
6.
2
2x – 3
6
2x – 3
2
3
+ 2
2 –
–
1
6
2
3
1
2+
1.
1
2
– +
5
6
2
3
1
4
–
2.
3
4
3
2
+
F. Combine and simplify the answers.
7.
2
x – 1
–
+
3
x + 3
x
x + 3
x
x – 1
8.
3
x + 2
–
+
3
x + 2
x
x – 2
x
x – 2
9.
2
x + h
–
2
x
h
10.
3
x – h
–
3
x
h
11.
2
x + h – 2
x – h
h
12.
3
x + h
–
h
3
x – h

123. G. Rationalize the denominator.
1.
1 – 3
1 + 3
2.
5 + 2
3 – 2
3.
1 – 33
2 + 3
4.
1 – 53
4 + 23
5.
32 – 33
22 – 43
6.
25 + 22
34 – 32
7.
42 – 37
22 – 27
8.
x + 3
x – 3
9. 3x – 3
3x + 2
10. x – 2
x + 2 + 2
11. x – 4
x – 3 – 1
Algebra of Radicals

124. (Answers to odd problems) Exercise A.
Applications of Factoring
1. (x + 1)(x – 4), 6, – 4, 6 3. (x + 1)(x – 2), – 9/4, 4, – 5/4
Exercise B.
1. positive, negative, negative, positive
3. negative, positive, negative, positive
5. x2(x2 – 3), – 2, –2, 550 7. , –3/5, 7/3
5. positive, negative, positive
Exercise C.
1. 4
x2
12
5x33. 5. 7. 3(x – 3)2
25(3x – 1) 2
5
3
(x + 2)(x – 2)
x+4
9. x (x – 2)
x2 – 1
11. x – 2
x + 2
13. x3 + x + 2
x (–3 x2 + x + 2)
15. x (x + 2)(x2 + 2)
(x – 2) 2

125. Multiplication and Division of Rational Expressions
Exercise D.
1. (x + 3)(x – 1) 3. 5.–(x + 1)(2x + 3) (2x – 3)(3x + 2)
7. 3x2 – 12x – 1 9. – 5 11. – 3
Exercise E.
3.
7(x + 1)
2(3x – 2)
5.
x + 3
1 – 2x
1. 7.
x2 + 9
9 – x2
4(x + 2)
3x – 2
34x2 – 9x – 20
5(2x – 5)(2x – 1)
9. 2(x2 + 3x + 4)
(x – 2) 2(x + 2)
11.
x2 + 3x – 3
(x2 – 9)(x + 2)
13. x2 + 8x – 12
x(x – 1)(x + 3)
15.

126. Complex Fractions
– 4x + 1
6x + 2
3. (6x-17)(x+4)5.
14(2x+3)(x+1)
15
11
1.
Exercise F.
x2 + x + 67.
x2 + 3
– 2
x(h + x)
9. 4
(h2 - x2)
11.
Exercise G.
1. 3 – 2 3. 11 – 73 5.
3
20
(4 – 6) 13 – 14
10
7.
– 9x + 15x – 69.
4 – 9x
13. 1
x + h + x
(x – 4)(x+2 – 2)11.
x – 4
15. 5
5x+5h+1 + 5x+1