# Finding Roots

**Mini Lecture 9.1**

Finding Roots

**Learning Objectives:**

1. Find square roots.

2. Evaluate models containing square roots.

3. Use a calculator to find decimal approximations for irrational square roots.

4. Find higher roots.

**Examples:**

Evaluate.

Given the equation, ,
solve for y given that :

Use a calculator to approximate each expression and round
to three decimal places. If the

expression is not a real number, so state.

Find the indicated root, or state that the expression is
not a real number.

**Teaching Notes:**

• The symbol is called the radical sign .

• The number under the radical sign is called the radicand.

• Together we refer to the radical sign and its radicand as a radical.

• The symbol - is used to denote the negative
square root of a number.

• The square root of a negative number is not a real number. This also applies
to any even

root .

• Not all radicals are square roots.

Answers:

c. 3.162 d. not a real number 4. a. 3 b. -1 c. not a real number d. -2 e. 2 f.
-4 g. -1

h. -2

**Mini Lecture 9.2**

Multiplying and Dividing Radicals

**Learning Objectives:**

1. Multiply square roots.

2. Simplify square roots .

3. Use the quotient rule for square roots .

4. Use the product and quotient rules for other roots.

**Examples:**

1. Multiply using the product rule.

2. Simplify using the product rule .

3. Simplify. (Look for a pattern) Assume all variables
represent positive number only.

4. Simplify.

5. Multiply. Then simplify if possible.

6. Simplify.

7. Simplify.

**Teaching Notes:**

• Have students memorize perfect square numbers through 225 and perfect
cubes through 216.

• Get as much out of the radicand as possible.

• Since radicals are unfamiliar to most students, it is important they see the
relationship of

squaring numbers and square roots, cubing numbers and cube roots, etc.

Answers:

**Mini Lecture 9.3**

Operations with Radicals

**Learning Objectives:**

1. Add and subtract radicals.

2. Multiply radical expressions with more than one term.

3. Multiply conjugates.

**Examples:**

Add or subtract as indicated.

Multiply.

**Teaching Notes:**

• Two or more square roots can be combined using the distributive property
provided they

have the same radicand.

• In some cases, radicals can be combined after they have been simplified.

• When multiplying radical expressions, distribute. This is similar to
multiplying a

monomial by a polynomial.

• When multiplying radical expressions use the FOIL method like multiplying
binomials.

• When multiplying conjugates (expressions that involve the sum and difference
of the

same two terms ), the FOIL method may be used or the special product formula.
When

using the FOIL method with conjugates the OI (outside & inside) will equal 0.

Answers: e. cannot be combined

**Mini Lecture 9.4**

Rationalizing the Denominator

**Learning Objectives:**

1. Rationalize denominators containing one term.

2. Rationalize denominators containing two terms.

**Examples:**

Multiply and simplify.

Rationalize each denominator.

State the conjugate of each of the following.

Multiply.

Rationalize each denominator and write in simplest form.

**Teaching Notes:**

• Remind students of the definition of a rational number. This will help
them understand

the meaning of “rationalizing the denominator.”

• It may be helpful to discuss the special product of (a + b)(a - b) with
several examples to

let students “see” again what happens to the middle term when the binomials are
foiled.

Answers:

**Mini Lecture 9.5**

Radical Equations

**Learning Objectives:**

1. Solve radical equations.

2. Solve problems involving square root models.

**Examples:**

Solve each radical equation. If the equation has no solution , so state.

**Teaching Notes:**

• A radical equation is an equation in which the variable occurs in a square
root, cube root,

or any higher root.

• To solve a radical equation containing square roots, first arrange terms so
that one radical

is isolated on one side of the equation. Next, square both sides of the equation
to

eliminate the square root. Solve and ALWAYS check the answer in the original
equation .

• There may be an extra solution(s) that does not check in the original
equation. This

solution(s) is/are called extraneous solutions.

Answers: 1. a. x = 2 b. x = 5 c. x = 3 d. no solution e. -3 f. no solution g. 10
h. no solution

i. 81 j. -2, -1

**Mini Lecture 9.6**

Rational Exponents

**Learning Objectives:**

1. Evaluate expressions with rational exponents.

2. Solve problems using models with rational exponents.

**Examples:**

Write each of the following in radical form first, then simplify.

Simplify.

**Teaching Notes:**

• If a graphing calculator is being used in the class, it is **helpful**
to show that is the same

as the using number values.

• Stress to students that the denominator of a rational exponent is the index of
the

corresponding radical expression.

• When the numerator of a rational exponent is not 1, the numerator is the power
to which

the radical is raised. It is usually easier to simplify it this way, but it is
possible to raise

the radicand to the power instead.

• When the exponent is negative , write the base as its reciprocal, and raise to
the positive

power.

Answers:

# Math Objectives

After successful completion of MAT 1010, a student should
be able to:

1. Solve a linear equation .

2. Use a mathematical model or formula to solve an application problem.

3. Solve a linear inequality and sketch the graph .

4. Solve a compound inequality.

5. Solve an application problem involving inequalities .

6. Solve an absolute value equation.

7. Solve an absolute value inequality.

8. Plot points on a rectangular coordinate system.

9. Use the Distance Formula to determine the distance between two points.

10. Sketch the graph of an equation using the point-plotting method.

11. Find and use the x-intercepts and y-intercepts as aids to sketching a graph.

12. Determine the slope of a line through two points.

13. Write the equation of a line .

14. Identify the domain and range of a relation or function.

15. Use function notation and evaluate a function.

16. Sketch the graph of a function on a rectangular coordinate system.

17. Use the Vertical Line Test to determine if a graph represents a function.

18. Identify transformations of the graph of a function and sketch their graphs.

19. Identify the leading coefficient and the degree of a polynomial.

20. Add and subtract polynomials.

21. Use the rules for exponents to simplify an expression.

22. Multiply polynomials.

23. Use special product formulas to multiply two binomials.

24. Factor the greatest common monomial factor from a
polynomial.

25. Factor a polynomial by grouping terms.

26. Factor the difference of two squares and factor the sum and difference of
two

cubes.

27. Factor polynomials completely by repeated factoring.

28. Factor trinomials.

29. Solve a polynomial equation by factoring .

30. Simplify an exponential expression involving negative exponents.

31. Write a very large or a very small number in scientific notation.

32. Simplify a rational expression.

33. Multiply and divide rational expressions and simplify .

34. Add and subtract rational expressions and simplify.

35. Simplify complex fractions.

36. Divide a polynomial by a monomial and write in simplest form.

37. Use long division to divide a polynomial by a second polynomial.

38. Use synthetic division to divide a polynomial by a polynomial of the form
x-k.

39. Solve rational equations.

40. Solve an application problem involving a rational equation.

41. Determine the nth root of a number and evaluate a radical expression.

42. Use the rules of exponents to evaluate an expression with a rational
exponent.

43. Use the Multiplication and Division Properties of Radicals to simplify a
radical

expression.

44. Use the Distributive Property to add and subtract like radicals.

45. Use the Pythagorean Theorem in an application problem.

46. Multiply radical expressions.

47. Simplify a quotient involving radicals by rationalizing the denominator.

48. Solve a radical equation.

49. Solve an application problem involving a radical
equation.

50. Write the square root of a negative number in i-form and perform operations
on

numbers in i-form.

51. Add, subtract, multiply, and divide complex numbers.

52. Solve a quadratic equation by extracting roots.

53. Solve a quadratic equation by completing the square.

54. Use the Quadratic Formula to solve a quadratic equation .

55. Determine the type of solutions to a quadratic equation using the
discriminant .

56. Use a quadratic equation to solve an application problem.

57. Use test intervals to solve a quadratic inequality.

58. Use test intervals to solve a rational inequality .

59. Sketch the graph of a linear inequality in two variables.

60. Find the equation of a parabola and sketch its graph. Find the equation of a

circle and sketch its graph. (see below)

61. Find the equation of an ellipse and sketch its graph. (see below)

62. Find the equation of a hyperbola and sketch its graph. (see below)

63. Use horizontal and vertical asymptotes and a table of values to sketch the
graph

of a rational function.

64. Solve a system of equations graphically.

65. Solve a system of equations using the method of substitution.

66. Solve a system of equations using the method of elimination.

67. Solve an application problem using a system of equations.

68. Solve a linear system in three variables.

69. Evaluate and graph exponential functions.

70. Use the Horizontal Line Test to determine if a function has an inverse.

71. Find the inverse of a function.

72. Evaluate and graph a logarithmic function.

# Course Syllabus for Beginning and Intermediate Algebra

**Prerequisite: **A grade of C or better in Mathematics
98 or appropriate placement test

score or consent of department chairperson.

**Text:** Beginning and Intermediate Algebra, 2nd Edition, by
Miller/O'Neil/Hyde, McGraw

Hill, 2008.

**Course Description:** Solution of linear and absolute value equations and
linear

inequalities; integer and rational exponents, simplification of radicals; slope
and

graphing linear equations ; systems of linear equations; solution of quadratic
equations

by factoring, completing the square and using the quadratic formula ;
introduction to

functions; applications included throughout the course. Writing assignments, as

appropriate to the discipline, are part of the course.

Other materials: Students are required to have a scientific or a graphing
calculator.

You are not allowed to use a cell phone as a calculator.

**Internet Resources: **The textbook we are using is bundled with MathZone
software .

MathZone is a complete online tutorial and course management system for

mathematics and statistics, designed for greater ease of use than any other
system

available. MathZone is a powerful Web-based tutorial for homework, quizzing,
testing,

and multimedia instruction. Also available in CD-Rom format, MathZone offers:

• ** Practice Exercises ** based on the text and generated in unlimited
quantity for

as much practice as needed to master any objective.

• **Video clips** of classroom instructors showing how to solve
exercises from text

step by step.

• **e-Professor** animations that take the student through step-by-step
instructions,

delivered on-screen and narrated by a teacher on audio, for solving exercises

**Weekly Schedule:**

1. 1.[Jun 2-Jun 6]: Introduction to Geometry(R.3), Linear Equations in One
Variable

( Review B )[Optional], Linear Equations in Two Variables (Review C:C2-C5 only),

Slope-Intercept Form of a Line (3.4), Point-Slope Formula (3.5), Applications of

Linear Equation (3.6), Systems of Linear Equations in 2 variables (Review D: D1,

and D2 only)[Optional].

2. [Jun 9-Jun 13]: Solving Systems of Linear Equations by the Addition

Method(4.3), Applications of Linear Equations in Two Variables (4.4), Systems of

Linear Equations in Three Variables (9.1), Applications of Systems of Linear

Equations in Three variables (9.2), Polynomials and Properties of

Exponents(Review E), Factoring Polynomials and Solving Quadratic

Equations(Review F), Rational Expressions(Review G: G1,G2, and G3 only).

3. [Jun 16-Jun 20]: Least Common Denominator(7.3), Addition and Subtraction of

Rational Expressions(7.4), Complex Fractions(7.5), Rational Equations(7.6),

Applications of Rational Equations and Proportions(7.7).

4. [Jun 23-Jun 27]: Introductions to Relations(8.1), Introductions to
Functions(8.2),

Graphs of Functions(8.3), Compound Inequalities(10.1). **[Review & Exam I].**

5. [Jun 30-Jul 4]: Absolute Value Equations(10.3), Linear Inequalities in Two

Variables (10.5), Definition of the nth Root(11.1), Rational Exponents(11.2).

6. [Jul 7-Jul 11]: Simplifying Radical Expressions(11.3), Addition and
Subtraction of

Radicals(11.4), Multiplications of Radicals (11.5), Rationalization(11.6).

7. [Jul 14-Jul 18]: Radical Equations(11.7), Complex Numbers(11.8), Square Root

Property and Completing the Square(12.1), Quadratic Formula (12.2).

**[Review & Exam II].**

8. [Jul 21-Jul 25]: Equations in Quadratic Form(12.3), Graphs of Quadratic

Functions(12.4), Vertex of Parabola and Applications(12.5). **[Review & Final
**

Exam].