1. Chapter 11
Polynomials

2. Add & Subtract Polynomials
11-1
Add & Subtract Polynomials

3. Monomial
A constant, a variable, or a product of a constant and one or more variables
u (1/3)m s2t3

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

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

6. Polynomial Expressions with several terms that follow patterns.
4x3 + 3x2 + 15x + 2
3b2 – 2b + 4

7. Coefficient The constant (or numerical) factor in a monomial
3m coefficient = 3
u coefficient = 1
-s2t coefficient = -1

8. Like Terms
Terms that are identical or that differ only in their coefficients
Are 2x and 2y similar?
Are -3x2 and 2x2 similar?

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

10. 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

11. Are they Simplified? 2x2 – 5 + 4x + x2 3x + 4x – 5

12. 11-2
Multiply by a Monomial

13. Examples
(5a)(-3b)
3v2(v2 + v + 1)
12(a2 + 3ab2 – 3b3 – 10)

14. Divide and Find Factors
11-3
Divide and Find Factors

15. GREATEST COMMON FACTOR
The greatest integer that is a factor of all the given integers.

16. 2,3,5,7,11,13,17,19,23,29
Prime number - is an integer greater than 1 that has no positive integral factor other than itself and 1.

17. GREATEST COMMON FACTOR
Find the GCF of 25 and 100
25 = 5 x 5
100 = 2 x 2 x 5 x 5
GCF = 5 x 5 = 25

18. GREATEST COMMON FACTOR
Find the GCF of 12 and 36
12 =
36 =
GCF =

19. GREATEST COMMON FACTOR
Find the GCF of 14,49 and 56
14 =
49 =
56 =
GCF =

20. Factoring Polynomials
vw + wx
= w(v + x)

21. Factoring Polynomials
21x2 – 35y2 =

22. Factoring Polynomials
13e – 39ef =

23. Dividing Polynomials by Monomials5
= 5(m+ 7)÷5
= m + 7

24. Dividing Polynomials by Monomials
7x + 14 7
= 7x + 14
= x + 2

25. Dividing Polynomials by Monomials
6a + 8b 2
= 2(a +4b) ÷ 2
= a + 2b

26. Dividing Polynomials by Monomials
2x + 6x2 2x

27. Multiply Two Binomials
11-4
Multiply Two Binomials

28. Multiplying Binomials
When multiplying two binomials both terms of each binomial must be multiplied by the other two terms

29. Multiplying binomials
Using the F.O.I.L method helps you remember the steps when multiplying

30. 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

31. Example: (2a – b)(3a + 5b) F – 2a · 3a O – 2a · 5b I – (-b) ▪ 3a
L - (-b) ▪ 5b

32. Example: (x + 6)(x +4)
F – x ▪ x
O – x ▪ 4
I – 6 ▪ x
L – 6 ▪ 4

33. Find Binomial Factors in a Polynomial
11-5
Find Binomial Factors in a Polynomial

34. Procedure
Group the terms in the polynomial as pairs that share a common monomial factor
Extract the monomial factor from each pair

35. Procedure
If the binomials that remain for each pair are identical, write this as a binomial factor of the whole expression
The monomials you extracted create a second polynomial. This is the paired factor for the original expression

36. Example 4x3 + 4x2y2 + xy + y3 Group (4x3 + 4x2y2) and factor
Group (xy + y3) and factor
4x2(x +y2) + y(x + y2)
Answer: (x +y2) (4x2 + y)

37. Example 2x3 - 2x2y - 3xy2 + 3y3+ xz2 – yz2
Group (2x3 - 2x2y2 ) and factor
Group (- 3xy2 + 3y3) and factor
Group (xz2 – yz2) and factor Answer:

38. Special Factoring Patterns
11-6
Special Factoring Patterns

39. 11-6 Difference of Squares
(a + b)(a – b)= a2 - b2
(x + 5) (x – 5) = x2 - 25

40. 11-6 Squares of Binomials (a + b)2 = a2 + 2ab + b2
Also known as Perfect square trinomials

41. Examples
(x + 3)2 = ?
(y - 2)2 = ?
(s + 6)2 = ?

42. 11-7
Factor Trinomials

43. Factoring Pattern for x2 + bx + c, c positive
x2 + 8x + 15 = (x + 3) (x + 5)
Middle term is the sum of 3 and 5
Last term is the product of 3 and 5

44. Example y2 + 14y + 40 = (y + 10) (y + 4)
Middle term is the sum of 10 and 4
Last term is the product of 10 and 4

45. Example y2 – 11y + 18 = (y - 2) (y - 9)
Middle term is the sum of -2 and -9
Last term is the product of -2 and -9

46. Factoring Pattern for x2 + bx + c, c negative
x2 - x - 20 = (x + 4) (x - 5)
Middle term is the sum of 4 and -5
Last term is the product of 4 and - 5

47. Example y2 + 6y - 40 = (y + 10) (y - 4)
Middle term is the sum of 10 and -4
Last term is the product of 10 and - 4

48. Example y2 – 7y - 18 = (y + 2) (y - 9)
Middle term is the sum of 2 and -9
Last term is the product of 2 and -9

49. More on Factoring Trinomials
11-9
More on Factoring Trinomials

50. 11-9 Factoring Pattern for ax2 + bx + c
Multiply a(c) = ac
List the factors of ac
Identify the factors that add to b
Rewrite problem and factor by grouping

51. Example 2x2 + 7x – 9 List factors: (-2)(9) = -18
Factors: (-2)(9) add to 7
(2x2 -2x) + (9x – 9)
2x(x -1) + 9(x – 1)
(x-1)(2x +9)

52. Example 14x2 - 17x + 5 List factors: (14)(5) = 70
Factors: (-7)(-10) add to -17
14x2 -7x – 10x + 5
(14x2 – 7x) + (-10x +5)
7x(2x-1)- 5(2x -1)
(7x -5)(2x – 1)

53. Example 3x2 - 11x - 4 List factors: (-12)(1) = -12
Factors: (-12)(1) add to -11
3x2 -12x + 1x - 4
(3x2 – 12x) + (1x -4)
3x(x-4) + 1(1x -4)
(x -4)(3x + 1)

54. END
END