1. Products and Factors of Polynomials
Chapter 4
Products and Factors of Polynomials

2. Section 4-1
Polynomials

3. Constant -
A number
-2, 3/5, 0

4. Monomial-
A constant, a variable, or a product of a constant and one or more variables
u (1/3)m s2t x

5. Coefficient- The constant (or numerical) factor in a monomial
3m coefficient = 3
u coefficient = 1
- s2t3 coefficient = -1

6. Degree of a Variable-
The number of times the variable occurs as a factor in the monomial
For Example – 6xy3
What is the degree of x? y?

7. Degree of a monomial-
The sum of the degrees of the variables in the monomial. A nonzero constant has degree 0.
The constant 0 has no degree.

8. Examples-
6xy degree = 4
-s2t degree = 5
u degree = 1
degree = 0

9. Similar Monomials-
Monomials that are identical or that differ only in their coefficients
Also called like terms
Are - s2t3 and 2s2t3 similar?

10. Polynomial- A monomial or a sum of monomials.
The monomials in a polynomial are called the terms of the polynomial.

11. Examples-
x2 + (-4)x + 5
x2 – 4x + 5
What are the terms?
x2, -4x, and 5

12. Simplified Polynomial-
A polynomial in which no two terms are similar.
The terms are usually arranged in order of decreasing degree of one of the variables

13. Are they Simplified? 2x3 – 5 + 4x + x3 3x3 + 4x – 5

14. Degree of a Polynomial-
The greatest of the degrees of its terms after it has been simplified
What is the degree?
x4 + 3x x3 + 3x – 7
x – 5x x + 1
x4 – 2x2y3 + 6y -11

15. Adding Polynomials (x2 + 4x – 3) + (x3 – 2x2 + 6x – 7)
To add two or more polynomials, write their sum and then simplify by combining like terms
Add the following-
(x2 + 4x – 3) +
(x3 – 2x2 + 6x – 7)

16. Subtracting Polynomials
To subtract one polynomial from another, add the opposite of each term of the polynomial you’re subtracting
(x3 – 5x2 + 2x – 5) –
(2x2 – 3x + 5)

17. Using Laws of Exponents
Section 4-2
Using Laws of Exponents

18. Laws of Exponents
Let a and b be real numbers and m and n be positive integers in all the following laws

19. Law 1
am · an = am+n
x2 · x4 = x6
y3 · y5 = ?
m · m4 = ?

20. Law 2
(ab)m = ambm
(xy)3 = x3y3
(3st)2 = ?
(xy)5 = ?

21. Law 3
(am)n = amn
(x3)2 = x6
(x2y3)4 = ?
(2mn2)3 = ?

22. Using Distributive Law
Distribute the variable using exponent laws
3t2(t3 – 2t2 + t – 4) = ?
– 2x2(x3 – 3x + 4) = ?

23. Multiplying Polynomials
Section 4-3
Multiplying Polynomials

24. Binomial A polynomial that has two terms 2x + 3 4x – 3y
3xy – z

25. Trinomial A polynomial that has three terms 2x2 – 3x + 1 14 + 32z – 3x
mn – m2 + n2

26. Multiplying binomials
When multiplying two binomials both terms of each binomial must be multiplied by the other two terms
Using the F.O.I.L method helps you remember the steps when multiplying

27. F.O.I.L. Method F – multiply First terms O – multiply Outer terms
I – multiply Inner terms
L – multiply Last terms
Add all terms to get product

28. Binomial A polynomial that has two terms 2x + 3 4x – 3y
3xy – z

29. Trinomial A polynomial that has three terms 2x2 – 3x + 1 14 + 32z – 3x
mn – m2 + n2

30. Multiplying binomials
When multiplying two binomials both terms of each binomial must be multiplied by the other two terms
Using the F.O.I.L method helps you remember the steps when multiplying

31. F.O.I.L. Method F – multiply First terms O – multiply Outer terms
I – multiply Inner terms
L – multiply Last terms
Add all terms to get product

32. Example - (2a – b)(3a + 5b) F – 2a · 3a O – 2a · 5b I – (-b) ▪ 3a
L - (-b) ▪ 5b
6a2 + 10ab – 3ab – 5b2
6a2 + 7ab – 5b2

33. Example – (x + 6)(x +4) F – x ▪ x O – x ▪ 4 I – 6 ▪ x L – 6 ▪ 4
x2 + 4x + 6x + 24
x2 + 10x + 24

34. Special Products (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 - 2ab + b2
(a + b)(a – b) = a2 - b2

35. Using Prime Factorization
Section 4-4
Using Prime Factorization

36. Factor
A number over a set of numbers, you write it as a product of numbers chosen from that set
The set is called a factor set

37. Example The number 15 can be factored in the following ways
(1)(15) (-1)(-15)
(5)(3) (-3)(-5)

38. Prime Number
An integer greater than 1 whose only positive integral factors are itself and 1

39. Prime Factorization
If the factor set is restricted to the set of primes
To find it you write the integer as a product of primes

40. Example
350 = 2 x 175
= 2 x 5 x 35
= 2 x 5 x 5 x 7
So the prime factorization of 350 is 2 x 52 x 7

41. Greatest Common Factor
The greatest integer that is a factor of each number.
To find the GCF, take the least power of each common prime factor.

42. Example
What is the GCF of 100, 120, and 90? 10

43. Least Common Multiple
The least positive integer having each as a factor
To find the LCM, take the greatest power of each common prime factor.

44. Example
What is the LCM of 100, 120, and 90?
1800

45. Summary GCF – take the least power of each common prime factor.
LCM – take the greatest power of each prime factor

46. Factoring Polynomials
Section 4-5
Factoring Polynomials

47. Factor
To factor a polynomial you express it as a product of other polynomials
We will factor using polynomials with integral coefficients

48. Greatest Monomial Factor
The GCF of the terms
What is the GCF of
2x4 – 4x3 + 8x2?
2x2

49. Now factor:
2x4 – 4x3 + 8x2
Factor out 2x2
2x2(x2 – 2x + 4)

50. Perfect Square Trinomials
The polynomials in the form of a2 + 2ab + b2 and a2 – 2ab + b2 are the result of squaring a + b and a – b respectively

51. Difference of Squares
The polynomial a2 – b2 is the product of a + b and a - b

52. Factor Each Polynomial
z2 + 6z + 9
4s2 – 4 st + t2
25x2 – 16a2

53. Factored Form:
(z + 3)2
(2s – t)2
(5x + 4a)(5x – 4a)

54. Sum and Difference of Cubes
a3 + b3 =
(a + b)(a2 - ab + b2)
a3 – b3 =
(a – b)(a2 + ab + b2)

55. Factor Each Polynomial
8u3 + v3

56. Factor by Grouping
Factor each polynomial by grouping terms that have a common factor
Then factor out the common factor and write the polynomial as a product of two factors

57. Factor each Polynomial
3xy x + 2y
xy + 3y + 2x + 6

58. Factoring Quadratic Polynomials
Section 4-6
Factoring Quadratic Polynomials

59. Quadratic Polynomials
Polynomials of the form ax2 + bx + c
Also called second- degree polynomials

60. Terms
ax2 - quadratic term
bx - linear term
c - constant term

61. Quadratic Trinomial
A quadratic polynomial for which a, b, and c are all nonzero integers

62. Factoring Quadratic Trinomials
ax2 + bx + c can be factored into the form
(px + q)(rx + s) where p, q, r, and s are integers

63. Factors
a = pr
b = ps + qr
c = qs

64. Factor the Polynomial
x2 + 2x - 15
a = 1, so pr = 1
c = -15, so qs = -15
b = 2, so ps + qr = 2

65. Factor the Polynomials
15t2 - 16t + 4
3 - 2z - z2
x2 + 4x - 3

66. Irreducible
If a polynomial has more than one term and cannot be expressed as a product of polynomials of lower degree taken from a given factor set, it is irreducible
x2 + 4x - 3 is irreducible

67. Factored Completely
A polynomial is factored completely when it is written as a product of factors and each factor is either a monomial, a prime polynomial, or a power of a prime polynomial

68. Greatest Common Factor
The GCF of two or more polynomials is the common factor having the greatest degree and the greatest constant factor

69. Least Common Multiple
The LCM of two or more polynomials is the common multiple having the least degree and least positive constant factor

70. Solving Polynomial Equations
Section 4-7
Solving Polynomial Equations

71. Polynomial Equation
An equation that is equivalent to one with a polynomial as one side and 0 as the other
x2 = 5x + 24

72. Root The value of a variable that satisfies the equation
Also called the solution

73. Solving a polynomial Equation
You can factor the polynomial to solve the equation

74. Steps to Solving a polynomial Equation
Write the equation with 0 as one side
Factor the other side of the equation
Solve the equation obtained by setting each factor equal to 0

75. Example 1 Solve (x – 5)(x + 2) = 0 Step 1: already = 0
Step 2: already factored
Step 3: set each factor = 0
x - 5 = 0 x + 2 = 0
x = 5 x = -2

76. Example 2 Solve x2 = x + 30 1: x2 - x – 30 = 0 2: (x – 6)(x + 5) = 0
The solution set is {6, -5}

77. Zeros A number r is a zero of a function f if f(r) = 0
You can find zeros using the same method that is used to solve polynomial equations

78. Example Find the zeros of f(x) = (x – 4)3 – 4(3x – 16) 1: simplify
2: factor
3: set each factor = 0

79. Double Zero
A number that occurs as a zero of a function twice

80. Double Root
A number that occurs twice as a root of a polynomial equation

81. Solve
x = 10x
12 + 4m = m2

82. Problem Solving Using Polynomial Equations
Section 4-8
Problem Solving Using Polynomial Equations

83. Example 1
A graphic artist is designing a poster that consists of a rectangular print with a uniform border. The print is to be twice as tall as it is wide, and the border is to be 3 in. wide. If the area of the poster is to be 680 in2, find the dimensions of the print.

84. Solution Draw a diagram Let w = width and 2w = height
The dimensions are 6 in. greater than the print, so they are w + 6 and 2w + 6

85. Solution 4. The area is represented by (w + 6)(2w + 6) = 680
5. Solve the equation.

86. Example 2
The sum of two numbers is 9. The sum of their squares is Find the numbers.

87. Solution Let x = one number Then 9 – x = the other number
x2 + (9 – x)2 = 101

88. Solving Polynomial Inequalities
Section 4-9
Solving Polynomial Inequalities

89. Polynomial Inequality
An inequality that is equivalent to an inequality with a polynomial as one side and 0 as the other side.
x2 > x + 6

90. Solve by factoring
The product is positive if both factors are positive, or both factors are negative
The product is negative if the factors have opposite signs

91. Example 1 Solve and graph x2 – 1 > x + 5 x2 – x – 6 > 0
Both factors must be positive or negative

92. Example 2 Solve and graph 3t < 4 – t2 t2 + 3t – 4 < 0
The factors must have opposite signs