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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].

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