1. Intermediate Algebra 098A
Review of
Exponents & Factoring

2. 1.1 – Integer Exponents
For any real number b and any natural number n, the nth power of b is found by multiplying b as a factor n times.

3. Exponential Expression – an expression that involves exponents
Base – the number being multiplied
Exponent – the number of factors of the base.

4. Product Rule

5. Quotient Rule

6. Integer Exponent

7. Zero as an exponent

8. Calculator Key
Exponent Key

9. Sample problem

10. more exponents
Power to a Power

11. Product to a Power

12. Polynomials - Review
Addition
and
Subtraction

13. Determine the coefficient and degree of a monomial
Objective:
Determine the coefficient and degree of a monomial

14. Def: Monomial
An expression that is a constant or a product of a constant and variables that are raised to whole –number powers.
Ex: 4x xyz

15. Coefficient: The numerical factor in a monomial
Definitions:
Coefficient: The numerical factor in a monomial
Degree of a Monomial: The sum of the exponents of all variables in the monomial.

16. Examples – identify the degree

17. A monomial or an expression that can be written as a sum or monomials.
Def: Polynomial:
A monomial or an expression that can be written as a sum or monomials.

18. Def: Polynomial in one variable:
A polynomial in which every variable term has the same variable.

19. Binomial: A polynomial containing two terms.
Definitions:
Binomial: A polynomial containing two terms.
Trinomial: A polynomial containing three terms.

20. The greatest degree of any of the terms in the polynomial.
Degree of a Polynomial
The greatest degree of any of the terms in the polynomial.

21. Examples:

22. Objective
Add
and
Subtract
Polynomials

23. To add or subtract Polynomials
Combine Like Terms
May be done with columns or horizontally
When subtracting- change the sign and add

24. Evaluate Polynomial Functions
Use functional notation to give a polynomial a name such as p or q and use functional notation such as p(x)
Can use Calculator

25. Calculator Methods 1. Plug In 2. Use [Table] 3. Use program EVALUATE
4. Use [STO->]
5. Use [VARS] [Y=]
6. Use graph- [CAL][Value]

26. Apply evaluation of polynomials to real-life applications.
Objective:
Apply evaluation of polynomials to real-life applications.

27. Multiplication and Special Products
Intermediate Algebra 5.4
Multiplication
and
Special Products

28. Objective
Multiply a
polynomial
by a
monomial

29. Procedure: Multiply a polynomial by a monomial
Use the distributive property to multiply each term in the polynomial by the monomial.
Helpful to multiply the coefficients first, then the variables in alphabetical order.

30. Law of Exponents

31. Multiply Special Products.
Objectives:
Multiply Polynomials
Multiply Binomials.
Multiply Special Products.

32. Procedure: Multiplying Polynomials
1. Multiply every term in the first polynomial by every term in the second polynomial.
2. Combine like terms.
3. Can be done horizontally or vertically.

33. Multiplying Binomials
FOIL
First
Outer
Inner
Last

34. Product of the sum and difference of the same two terms Also called multiplying conjugates

35. Squaring a Binomial

36. Use techniques as part of a larger simplification problem.
Objective:
Simplify Expressions
Use techniques as part of a larger simplification problem.

37. Albert Einstein-Physicist
“In the middle of difficulty lies opportunity.”

38. Intermediate Algebra –098A
Common Factors
and
Grouping

39. A number or expression written as a product of factors.
Def: Factored Form
A number or expression written as a product of factors.

40. Greatest Common Factor (GCF)
Of two numbers a and b is the largest integer that is a factor of both a and b.

41. Can do two numbers – input with commas and ). Example: gcd(36,48)=12
Calculator and gcd
[MATH][NUM]gcd(
Can do two numbers – input with commas and ).
Example: gcd(36,48)=12

42. Greatest Common Factor (GCF) of a set of terms
Always do this FIRST!

43. Procedure: Determine greatest common factor GCF of 2 or more monomials
1. Determine GCF of numerical coefficients.
2. Determine the smallest exponent of each exponential factor whose base is common to the monomials. Write base with that exponent.
3. Product of 1 and 2 is GCF

44. Factoring Common Factor
1. Find the GCF of the terms
2. Factor each term with the GCF as one factor.
3. Apply distributive property to factor the polynomial

45. Example of Common Factor

46. Factoring when first terms is negative
Prefer the first term inside parentheses to be positive. Factor out the negative of the GCF.

47. Factoring when GCF is a polynomial

48. Factoring by Grouping – 4 terms
1. Check for a common factor
2. Group the terms so each group has a common factor.
3. Factor out the GCF in each group.
4. Factor out the common binomial factor – if none , rearrange polynomial
5. Check

49. Example – factor by grouping

50. Ralph Waldo Emerson – U.S. essayist, poet, philosopher
“We live in succession , in division, in parts, in particles.”

51. Intermediate Algebra 098A
Special Factoring

52. a difference of squares a perfect square trinomial a sum of cubes
Objectives:Factor
a difference of squares
a perfect square trinomial
a sum of cubes
a difference of cubes

53. Factor the Difference of two squares

54. Special Note
The sum of two squares is prime and cannot be factored.

55. Factoring Perfect Square Trinomials

56. Factor: Sum and Difference of cubes

57. Note
The following is not factorable

58. Factoring sum of Cubes - informal
(first + second)
(first squared minus first times second plus second squared)

59. Intermediate Algebra 098A
Factoring Trinomials of
General Quadratic

60. Objectives:
Factor trinomials of the form

61. Factoring
1. Find two numbers with a product equal to c and a sum equal to b.
The factored trinomial will have the form(x + ___ ) (x + ___ )
Where the second terms are the numbers found in step 1.
Factors could be combinations of positive or negative

62. Factoring Trial and Error
1. Look for a common factor
2. Determine a pair of coefficients of first terms whose product is a
3. Determine a pair of last terms whose product is c
4. Verify that the sum of factors yields b
5. Check with FOIL Redo

63. Factoring ac method
1. Determine common factor if any
2. Find two factors of ac whose sum is b
3. Write a 4-term polynomial in which by is written as the sum of two like terms whose coefficients are two factors determined.
4. Factor by grouping.

64. Example of ac method

65. Example of ac method

66. Factoring - overview 1. Common Factor 2. 4 terms – factor by grouping
3. 3 terms – possible perfect square
4. 2 terms –difference of squares
Sum of cubes
Difference of cubes
Check each term to see if completely factored

67. Isiah Thomas:
“I’ve always believed no matter how many shots I miss, I’m going to make the next one.”

68. Intermediate Algebra 098A
Solving Equations by
Factoring

69. If a and b are real numbers
Zero-Factor Theorem
If a and b are real numbers
and ab =0
Then a = 0 or b = 0

70. Example of zero factor property

71. Solving a polynomial equation by factoring.
Factor the polynomial completely.
Set each factor equal to 0
Solve each of resulting equations
Check solutions in original equation.
Write the equation in standard form.

72. Example – solve by factoring

73. Example: solve by factoring

74. Example: solve by factoring
A right triangle has a hypotenuse 9 ft longer than the base and another side 1 foot longer than the base. How long are the sides?
Hint: Draw a picture
Use the Pythagorean theorem

75. Solution
Answer: 20 ft, 21 ft, and 29 ft

76. Example – solve by factoring
Answer: {-1/2,4}

77. Example: solve by factoring
Answer: {-5/2,2}

78. Example: solve by factoring
Answer: {0,4/3}

79. Example: solve by factoring
Answer: {-3,-2,2}

80. Sugar Ray Robinson
“I’ve always believed that you can think positive just as well as you can think negative.”

82. Maya Angelou - poet
“Since time is the one immaterial object which we cannot influence – neither speed up nor slow down, add to nor diminish – it is an imponderably valuable gift.”