1. Chapter 0
Review of Algebra

2. INTRODUCTORY MATHEMATICAL ANALYSIS
0. Review of Algebra
Applications and More Algebra
Functions and Graphs
Lines, Parabolas, and Systems
Exponential and Logarithmic Functions
Mathematics of Finance
Matrix Algebra
Linear Programming
Introduction to Probability and Statistics

3. INTRODUCTORY MATHEMATICAL ANALYSIS
Additional Topics in Probability
Limits and Continuity
Differentiation
Additional Differentiation Topics
Curve Sketching
Integration
Methods and Applications of Integration
Continuous Random Variables
Multivariable Calculus

4. Chapter 0: Review of Algebra
Chapter Objectives
To be familiar with sets, real numbers, real-number line.
To relate properties of real numbers in terms of their operations.
To review the procedure of rationalizing the denominator.
To perform operations of algebraic expressions.
To state basic rules for factoring.
To rationalize the denominator of a fraction.
To solve linear equations.
To solve quadratic equations.

5. Chapter Outline 0.1) Sets of Real Numbers
Chapter 0: Review of Algebra
Chapter Outline
0.1)
Sets of Real Numbers
Some Properties of Real Numbers
Exponents and Radicals
Operations with Algebraic Expressions
Factoring
Fractions
Equations, in Particular Linear Equations
Quadratic Equations
0.2)
0.3)
0.4)
0.5)
0.6)
0.7)
0.8)

6. 0.1 Sets of Real Numbers A set is a collection of objects.
Chapter 0: Review of Algebra
0.1 Sets of Real Numbers
A set is a collection of objects.
An object in a set is called an element of that set.
Different type of integers:
The real-number line is shown as

7. 0.2 Some Properties of Real Numbers
Chapter 0: Review of Algebra
0.2 Some Properties of Real Numbers
Important properties of real numbers
The Transitive Property of Equality
The Closure Properties of Addition and Multiplication
The Commutative Properties of Addition and Multiplication

8. The Commutative Properties of Addition and Multiplication
Chapter 0: Review of Algebra
0.2 Some Properties of Real Numbers
The Commutative Properties of Addition and Multiplication
The Identity Properties
The Inverse Properties
The Distributive Properties

9. Example 3 – Applying Properties of Real Numbers Solution:
Chapter 0: Review of Algebra
0.2 Some Properties of Real Numbers
Example 1 – Applying Properties of Real Numbers
Example 3 – Applying Properties of Real Numbers
Solution:
Show that
Solution:

10. Show that Solution: Chapter 0: Review of Algebra
0.2 Some Properties of Real Numbers
Example 3 – Applying Properties of Real Numbers
Show that
Solution:

11. 0.3 Exponents and Radicals
Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Properties:
exponent
base

12. Example 1 – Exponents Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Example 1 – Exponents

13. The symbol is called a radical.
Chapter 0: Review of Algebra
0.3 Exponents and Radicals
The symbol is called a radical.
n is the index, x is the radicand, and is the radical sign.

14. Eliminate negative exponents in and simplify. Solution:
Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Example 3 – Rationalizing Denominators
Solution:
Example 5 – Exponents
Eliminate negative exponents in and simplify.
Solution:

15. b. Simplify by using the distributive law. Solution:
Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Example 5 – Exponents
b. Simplify by using the distributive law.
Solution:
Eliminate negative exponents in

16. d. Eliminate negative exponents in Solution:
Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Example 5 – Exponents
d. Eliminate negative exponents in
Solution:
e. Apply the distributive law to

17. a. Simplify Solution: Simplify Example 7 – Radicals
Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Example 7 – Radicals
a. Simplify
Solution:
Simplify

18. If x is any real number, simplify
Chapter 0: Review of Algebra
0.3 Exponents and Radicals
Example 7 – Radicals
c. Simplify
Solution:
If x is any real number, simplify
Thus, and

19. 0.4 Operations with Algebraic Expressions
Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions
If symbols are combined by any or all of the operations, the resulting expression is called an algebraic expression.
A polynomial in x is an algebraic expression of the form:
where n = non-negative integer cn = constants

20. is an algebraic expression in the variable x.
Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions
Example 1 – Algebraic Expressions
is an algebraic expression in the variable x.
is an algebraic expression in the variable y.
is an algebraic expression in the variables x and y.

21. Simplify Solution: Example 3 – Subtracting Algebraic Expressions
Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions
Example 3 – Subtracting Algebraic Expressions
Simplify
Solution:

22. A list of products may be obtained from the distributive property:
Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions
A list of products may be obtained from the distributive property:

23. By Rule 2, By Rule 3, Example 5 – Special Products
Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions
Example 5 – Special Products
By Rule 2,
By Rule 3,

24. c. By Rule 5, By Rule 6, By Rule 7, Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions
Example 5 – Special Products
c. By Rule 5,
By Rule 6,
By Rule 7,

25. Example 7 – Dividing a Multinomial by a Monomial
Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions
Example 7 – Dividing a Multinomial by a Monomial

26. Chapter 0: Review of Algebra
0.5 Factoring
If two or more expressions are multiplied together, the expressions are called the factors of the product.

27. Solution: b. Factor completely. Example 1 – Common Factors
Chapter 0: Review of Algebra
0.5 Factoring
Example 1 – Common Factors
a. Factor completely.
Solution:
b. Factor completely.

28. Chapter 0: Review of Algebra
0.5 Factoring
Example 3 – Factoring

29. 0.6 Fractions Simplifying Fractions
Chapter 0: Review of Algebra
0.6 Fractions
Simplifying Fractions
Allows us to multiply/divide the numerator and denominator by the same nonzero quantity.
Multiplication and Division of Fractions
The rule for multiplying and dividing is

30. Rationalizing the Denominator
Chapter 0: Review of Algebra
0.6 Fractions
Rationalizing the Denominator
For a denominator with square roots, it may be rationalized by multiplying an expression that makes the denominator a difference of two squares.
Addition and Subtraction of Fractions
If we add two fractions having the same denominator, we get a fraction whose denominator is the common denominator.

31. Solution: b. Simplify Example 1 – Simplifying Fractions a. Simplify
Chapter 0: Review of Algebra
0.6 Fractions
Example 1 – Simplifying Fractions
a. Simplify
Solution:
b. Simplify

32. Example 3 – Dividing Fractions
Chapter 0: Review of Algebra
0.6 Fractions
Example 3 – Dividing Fractions

33. Example 5 – Adding and Subtracting Fractions
Chapter 0: Review of Algebra
0.6 Fractions
Example 5 – Adding and Subtracting Fractions

34. Example 7 – Subtracting Fractions
Chapter 0: Review of Algebra
0.6 Fractions
Example 7 – Subtracting Fractions

35. 0.7 Equations, in Particular Linear Equations
Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Equations
An equation is a statement that two expressions are equal.
The two expressions that make up an equation are called its sides.
They are separated by the equality sign, =.

36. Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Example 1 – Examples of Equations
A variable (e.g. x, y) is a symbol that can be replaced by any one of a set of different numbers.

37. There are three operations that guarantee equivalence:
Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations
Equivalent Equations
Two equations are said to be equivalent if they have exactly the same solutions.
There are three operations that guarantee equivalence:
Adding/subtracting the same polynomial to/from both sides of an equation.
Multiplying/dividing both sides of an equation by the same nonzero constant.
Replacing either side of an equation by an equal expression.

38. Operations That May Not Produce Equivalent Equations
Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations
Operations That May Not Produce Equivalent
Equations
Multiplying both sides of an equation by an expression involving the variable.
Dividing both sides of an equation by an expression involving the variable.
Raising both sides of an equation to equal powers.

39. A linear equation in the variable x can be written in the form
Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations
Linear Equations
A linear equation in the variable x can be written in the form
where a and b are constants and
A linear equation is also called a first-degree equation or an equation of degree one.

40. Solve Solution: Example 3 – Solving a Linear Equation
Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Example 3 – Solving a Linear Equation
Solve
Solution:

41. Solve Solution: Example 5 – Solving a Linear Equations
Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Example 5 – Solving a Linear Equations
Solve
Solution:

42. The letters are called literal constants.
Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations
Literal Equations
Equations where constants are not specified, but are represented as a, b, c, d, etc. are called literal equations.
The letters are called literal constants.

43. Solve for x. Solution: Example 7 – Solving a Literal Equation
Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations
Example 7 – Solving a Literal Equation
Solve for x.
Solution:

44. Chapter 0: Review of Algebra 0
Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations
Example 9 – Solving a Fractional Equation
Solve
Solution:
Fractional Equations
A fractional equation is an equation in which an unknown is in a denominator.

45. If express u in terms of the remaining letters; that is, solve for u.
Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations
Example 11 – Literal Equation
If express u in terms of the remaining letters; that is, solve for u.
Solution:

46. A radical equation is one in which an unknown occurs in a radicand.
Chapter 0: Review of Algebra 0.7 Equations, in Particular Linear Equations
Radical Equations
A radical equation is one in which an unknown occurs in a radicand.
Example 13 – Solving a Radical Equation
Solve
Solution:

47. where a, b, and c are constants and
Chapter 0: Review of Algebra
0.8 Quadratic Equations
A quadratic equation in the variable x is an equation that can be written in the form
where a, b, and c are constants and
A quadratic equation is also called a second-degree equation or an equation of degree two.

48. Solution: Factor the left side factor:
Chapter 0: Review of Algebra 0.8 Quadratic Equations
Example 1 – Solving a Quadratic Equation by Factoring
a. Solve
Solution: Factor the left side factor:
Whenever the product of two or more quantities is zero, at least one of the quantities must be zero.

49. Chapter 0: Review of Algebra 0.8 Quadratic Equations
Example 1 – Solving a Quadratic Equation by Factoring
b. Solve
Solution:

50. a. Solve Solution: b. Solve
Chapter 0: Review of Algebra 0.8 Quadratic Equations
Example 3 – Solving a Higher-Degree Equation by Factoring
a. Solve
Solution:
b. Solve

51. Solve Solution: Example 5 – Solution by Factoring
Chapter 0: Review of Algebra 0.8 Quadratic Equations
Example 5 – Solution by Factoring
Solve
Solution:

52. The roots of the quadratic equation
Chapter 0: Review of Algebra
0.8 Quadratic Equations
Quadratic Formula
The roots of the quadratic equation
can be given as

53. Solve by the quadratic formula.
Chapter 0: Review of Algebra 0.8 Quadratic Equations
Example 7 – A Quadratic Equation with One Real Root
Solve by the quadratic formula.
Solution: Here a = 9, b = 6√2, and c = 2. The roots are

54. Quadratic-Form Equation
Chapter 0: Review of Algebra
0.8 Quadratic Equations
Quadratic-Form Equation
When a non-quadratic equation can be transformed into a quadratic equation by an appropriate substitution, the given equation is said to have quadratic-form.

55. This equation can be written as Substituting w =1/x3, we have
Chapter 0: Review of Algebra
0.8 Quadratic Equations
Example 9 – Solving a Quadratic-Form Equation
Solve
Solution:
This equation can be written as
Substituting w =1/x3, we have
Thus, the roots are