1. Topic 7: Polynomials

2. Table of Contents
1. Introduction to Polynomials 2. Adding & Subtracting Polynomials 3. Multiplying Polynomials 4. Factoring Polynomials 5. Factoring Polynomials, part 2 6. Solving Quadratics with Factoring 7. Factoring by Grouping 8. Completing the Square

3. Introduction to Polynomials

4. Vocab
Monomial: a number, a variable, or a product of numbers and variables with whole-number exponents.
Degree of a monomial: is the sum of the exponents of the variables. A constant has degree 0.

5. Example: Degree of a Monomial
Find the degree of each monomial. A.
4p4q3
B. 7ed
C. 3

6. Let’s Practice…. Find the degree of each monomial. a. 1.5k2m b. 4x c.

7. Review: Like Terms 4a3b2 + 3a2b3 – 2a3b2
You can add or subtract monomials by adding or subtracting like terms.
Like terms
The variables have the same powers.
4a3b2 + 3a2b3 – 2a3b2
Not like terms
The variables have different powers.

8. Identify Like Terms Identify the like terms in each polynomial.
A. 5x3 + y – 6y2 + 4x3
B. 3a3b2 + 3a2b3 + 2a3b2 – a3b2
Like terms: ______________________
Like terms: _______________________

9. Let’s Practice… Identify the like terms in each polynomial.
A. 4y4 + y2 + 2 – 8y2 + 2y4
B. 7n4r2 + 3n2r3 + 5n4r2 + n4r2
Like terms: ____________________________
Like terms: ___________________________

10. Add or Subtract Monomials
Simplify.
A. 4x2 + 2x2

11. Add or Subtract Monomials
Simplify.
B. 3n5m4 - n5m4

12. Let’s Practice…
Simplify.
2x3 - 5x3
2n5p4 + n5p4

13. Vocab Polynomial: an expression of more than two algebraic terms.
Example: 3x4 + 5x2 – 7x + 1
Degree of a polynomial is the degree of the term with the greatest power/exponent.
Example: The degree of 3x4 + 5x2 – 7x + 1 is 4.

14. Degree of a Polynomial A. 11x7 + 3x3
Find the degree of each polynomial.
A. 11x7 + 3x3 B.

15. Let’s Practice… a. 5x – 6 b. x3y2 + x2y3 – x4 + 2
Find the degree of each polynomial.
a. 5x – 6
b. x3y2 + x2y3 – x4 + 2

16. Vocab
Standard form of a polynomial: Polynomial written with the terms in order from greatest degree to least degree.
Leading Coefficient: When written in standard form, the coefficient of the first term is called the leading coefficient.
Example: 3x4 + 5x2 – 7x + 1 and 3 is the leading coefficient.

17. Let’s Practice… 1. 6x – 7x5 + 4x2 + 9 2. 16 – 4x2 + x5 + 9x3
Write the polynomial in standard form. Then give the leading coefficient.
1. 6x – 7x5 + 4x2 + 9
2. 16 – 4x2 + x5 + 9x3
3. 18y5 – 3y8 + 14y

18. Special Polynomial Names
By Degree
Degree
Name
By # of Terms 1 2
Constant
Linear
Quadratic 3 4 5
6 or more
6th,7th,degree and so on
Cubic
Quartic
Quintic
Name
Terms
Monomial
Binomial
Trinomial
Polynomial
4 or more 1 2 3

19. Name the Following… A. 5n3 + 4n B. 4y6 – 5y3 + 2y – 9 C. –2x
Classify each polynomial according to its degree and number of terms.
A. 5n3 + 4n
B. 4y6 – 5y3 + 2y – 9
C. –2x

20. Let’s Practice… a. x3 + x2 – x + 2 b. 6 c. –3y8 + 18y5 + 14y
Classify each polynomial according to its degree and number of terms.
a. x3 + x2 – x + 2
b. 6
c. –3y8 + 18y5 + 14y

21. Adding and Subtracting Polynomials

22. Adding and Subtracting Polynomials
Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms.
Remember!
Like terms are constants or terms with the same variable(s) raised to the same power(s).

23. Simplifying Polynomials
Combine like terms.
A. 12p3 + 11p2 + 8p3
B. 5x2 – 6 – 3x + 8

24. Let’s Practice… Combine like terms. a. 2s2 + 3s2 + s – 3s2 – 5s
b. 4z4 – z2 +16z z3 – 7

25. Let’s Practice… Combine like terms.
c. 2x8 + 7y8 – x8 – 9y7 - 10y7 + y8
d. 9b3c2 + -4b3 + 5c2 + 5b3c2 – 13b3c2

26. 2 Methods: Adding Polynomials
Polynomials can be added in either vertical or horizontal form.
In vertical form, align the like terms and add:
In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms.
5x2 + 4x + 1
+ 2x2 + 5x + 2
7x2 + 9x + 3
(5x2 + 4x + 1) + (2x2 + 5x + 2)
= (5x2 + 2x2 + 1) + (4x + 5x) + (1 + 2)
= 7x2 + 9x + 3

27. Adding Polynomials Add. A. (4m2 + 5) + (m2 – m + 6)
B. (10xy + x) + (–3xy + y)

28. Let’s Practice…
Add.
Add (5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a).

29. Subtracting Polynomials
To subtract polynomials, remember that subtracting is the same as adding the opposite (distributing the negative).
To find the opposite of a polynomial, you must write the opposite of each term in the polynomial:
–(2x3 – 3x + 7)= –2x3 + 3x – 7

30. Subtracting Polynomials
(–10x2 – 3x + 7) – (x2 – 9)

31. Subtracting Polynomials
(x3 + 4y) – (2x3)
(7m4 – 2m2) – (5m4 – 5m2 + 8)

32. Let’s Practice… Subtract. (9q2 – 3q) – (q2 – 5)
(2x2 – 3x2 + 1) – (x2 + x + 1)

33. Multiplying Polynomials F.O.I.L

34. Multiplying Polynomials
Each term in the first polynomial, must be multiplied by each term in the second polynomial.

35. Method 1: Distribute
First
Outer
Inner
Last
Multiply!!!

36. “F.O.I.L.”

37. Method 2: Box 3x -5 5x +2 Multiply (3x – 5)(5x + 2) Draw a box.
Write a polynomial on the top and side of a box.
Multiply.
Combine like terms. 3x -5 5x +2

38. Let’s Practice… 1. (7x – 10)(3x + 8) 2. (2x – 3)(4x - 8)

39. Multiplying Terms with Exponents
When FOILing, add the exponents and multiply coefficients.
Add the little numbers and multiply the big numbers!!!
Example:
(3x2 + 10x)(5x3 – 7x2)
15x5 - 21x4 + 50x4 – 70x3
15x5 + 29x4 – 70x3

40. Let’s Practice… 1. (7x2 – 10x)(3x3 + 8x2) 2. (2x4 – 3x2)(4x - 8)

41. Multiplying Larger Polynomials
Each term in the 1st polynomial must be multiplied by each term in the 2nd.
Example:
(7x2 + 2x + 8)(4x3 – 9x2)

42. Method 2: Multiply: (2x - 5)(x2 - 5x + 4)

43. Let’s Practice…
(5x2 + 7) (2x3 – 5x2 +9)
(10x4 – 5x2 + 8) (8x3 -3x -6)

44. Factoring Polynomials

45. Vocab: Factoring
Factoring is rewriting an expression as a product of factors.
It is the reverse of multiplying polynomials FOILing.

46. To determine the factors, ask yourself…
What two #’s add to the middle number AND multiply to the last number?!?!

47. Let’s Practice…
What adds (or subtracts) to get 3 and multiplies to get 2?
What adds (or subtracts) to get -7 and multiplies to get 10?
What adds (or subtracts) to get -7 and multiplies to get -44?

48. Let’s Practice…
Factor:
x2 + 5x + 6
2. x2 -7x + 10
3. x2 -11x +24

49. Signs of Factors b c Factors + +,+ -
+,- (The factor w/ the greater absolute value is -)
+,- (The factor w/ the greater absolute value is +)
-, -

50. Vocab: GCF
The greatest common factor (GCF) is a common factor of the terms in the expression. Example:

51. Vocab: Prime
If a polynomials is “prime” it means there are no factors. Find the prime polynomials below: 1. x2 + 7x x2 + 5x x2 + 9x + 10

52. Let’s Practice…. Factor. y2 -10y +16 2. r2 -11r +24 3. n2 -15n +56
4. v2 + 5v -36

54. Let’s Practice…
x2 + 12x + 36
x2 - 8x + 16
-x2 +11x -18
16- x2

55. Factoring Polynomials, Part 2

56. Expanded Form
Expanded Form
When factoring problems where a ≠ 1, we first want to get the problem into expanded form before we try to factor.

57. Creating Expanded Form
Step 1: Multiply a·c
Step 2: To get to expanded form ask yourself “What multiplies to get a·c, and add/subtracts to get to b.”
Example:
Expand: 2x2 +9x +7
Expand: 3x2 + 2x – 8

58. Method 1:
Step 3: Write your new factors in place of bx. Step 4: Group the first two terms together and the last two terms together. Step 5: Factor each group Step 6: Factor again to get the complete factorization

59. Method 1: 6x2 + 13 x +5 Multiply a·c (6·5=30)
To get to expanded form ask yourself “What multiplies to get a·c, and add/subtracts to get to b.” (10, 3)
Write your new factors in place of bx. (6x2+10x+3x+5)
Group the first two terms together and the last two terms together. [(6x2+10x)+(3x+5)]
Factor each group [2x(3x+5)+1(3x+5)]
Factor again to get the complete factorization [(3x+5)(2x+1)]

60. Method 2: 6x x +5
Original 1st Term
Expanded Term 1
Expanded Term 2
Original Last Term
Step 3: Fill in box. Step 4: Factor horizontally and vertically. Step 5: Terms outside of box are the solution.

61. Factor: 2x2 + 5x -12 Original 1st Term Expanded Term 1 Expanded Term 2
Original Last Term

62. Factor: 3x2 + 7x +2

63. Factor: 2x2 + 15x -8

64. Factor: 16x2 + 28x +10

65. Factoring by Grouping

66. Factoring by Grouping
Using the distributive property to factor polynomials with four or more terms.
Terms can be put into groups and then factored---- each group will have a “like” factor used in regrouping.

67. Factoring by Grouping
A polynomial can be factored by grouping if all of the following conditions exist.
There are four or more terms.
Terms have common factors that can be grouped together, and
There are two common factors that are identical.
Symbols:
ax + bx + ay + by = (ax + bx) + (ay + by)
Group, factor
Regroup
= x(a + b) + y(a + b)
= (x + y)(a + b)

68. Factor by Grouping 6h4 – 4h3 + 12h – 8
Factor each polynomial by grouping. Check your answer.
6h4 – 4h3 + 12h – 8

69. Factor by Grouping 5y4 – 15y3 + y2 – 3y
Factor each polynomial by grouping.
5y4 – 15y3 + y2 – 3y

70. Let’s Practice… Factor each polynomial by grouping.
6b3 + 8b2 + 9b + 12

71. Let’s Practice…
Factor each polynomial by grouping.
4r3 + 24r + r2 + 6

72. Factoring with Opposite Groups
2x3 – 12x – 3x

73. Let’s Practice… 15x2 – 10x3 + 8x – 12
Factor each polynomial. Check your answer.
15x2 – 10x3 + 8x – 12

74. Let’s Practice… Factor each polynomial by grouping.
1. 2x3 + x2 – 6x – 3
2. 7p4 – 2p p – 18

75. Factoring Procedure

76. Completing the Square

77. How to: Square root each side and solve.
Rearrange equation so it is in the form ax2 + bx = c
Divide every term on both sides by a.
Add ( 𝑏 2 )2 to both sides of the equation.
Factor.
Square root each side and solve.

78. Solve: x2 + 6x - 3 = 0

79. Solve: x2 - 12x + 7 = 0

80. Solve: 2x2 + 2x -5 = x2

81. Solve: 9x2 - 12x - 2 = 0