1. Polynomial Operations
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2. Polynomial Operations
In the last section, we introduced polynomial expressions.
#xN ± #xN-1 ± … ± #x ± #

3. Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms

4. Terms with the same variabe part are called like-terms.
Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms

5. Terms with the same variabe part are called like-terms.
Like-terms may be combined.
Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms

6. Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms

7. Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms

8. Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms

9. Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
In particular x2 + x2 = 2x2 = x4,
Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms

10. Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5.
Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms

11. Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5.
Note that we write 1xN as xN , –1xN as –xN.
Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms

12. Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it to the
coefficient.
Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms

13. Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it to the
coefficient. Hence, 3(5x) = (3*5)x =15x,
Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms

14. Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it to the
coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms

15. Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it to the
coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
When multiplying a number to a polynomial, we may expand
the result using the distributive law: A(B ± C) = AB ± AC.
Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms

16. Example A. Expand and simplify.
Polynomial Operations

17. Example A. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
Polynomial Operations

18. Example A. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12
Polynomial Operations

19. Example A. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
Polynomial Operations

20. Example A. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
Polynomial Operations

21. Example A. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
Polynomial Operations

22. Example A. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15
Polynomial Operations

23. Example A. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
Polynomial Operations

24. Example A. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomial Operations

25. Example A. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.

26. Example A. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example B.
a. (3x2)(2x3) =
b. 3x2(–4x) =
c. 3x2(2x3 – 4x)
=

27. Example A. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example B.
a. (3x2)(2x3) = 3*2x2x3
b. 3x2(–4x) =
c. 3x2(2x3 – 4x)
=

28. Example A. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example B.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) =
c. 3x2(2x3 – 4x)
=

29. Example A. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example B.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) = 3(–4)x2x = –12x3
c. 3x2(2x3 – 4x)
=

30. Example A. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example B.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) = 3(–4)x2x = –12x3
c. 3x2(2x3 – 4x) distribute
= 6x5 – 12x3

31. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations

32. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example C.
a. (3x + 2)(2x – 1)

33. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example C.
= 3x(2x – 1) + 2(2x – 1)
a. (3x + 2)(2x – 1)

34. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example C.
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
a. (3x + 2)(2x – 1)

35. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example C.
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
a. (3x + 2)(2x – 1)

36. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example C.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
a. (3x + 2)(2x – 1)

37. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example C.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
a. (3x + 2)(2x – 1)

38. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example C.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
a. (3x + 2)(2x – 1)

39. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example C.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
= 4x3 + 4x2 – 11x + 4
a. (3x + 2)(2x – 1)

40. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example C.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
= 4x3 + 4x2 – 11x + 4
a. (3x + 2)(2x – 1)
Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1)
instead, we get the same answers. (Check this.)

41. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example C.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
= 4x3 + 4x2 – 11x + 4
a. (3x + 2)(2x – 1)
Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1)
instead, we get the same answers. (Check this.)
Fact. If P and Q are two polynomials then PQ ≡ QP.

42. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example C.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
= 4x3 + 4x2 – 11x + 4
a. (3x + 2)(2x – 1)
Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1)
instead, we get the same answers. (Check this.)
Fact. If P and Q are two polynomials then PQ ≡ QP.
A shorter way to multiply is to bypass the 2nd step and use the
general distributive law.

43. General Distributive Rule:
Polynomial Operations

44. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
Polynomial Operations

45. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..
Polynomial Operations

46. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..
Polynomial Operations

47. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Polynomial Operations

48. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
Polynomial Operations

49. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
= x2
Polynomial Operations

50. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
= x2 – 4x
Polynomial Operations

51. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x
Polynomial Operations

52. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12
Polynomial Operations

53. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
Polynomial Operations

54. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations

55. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3

56. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2

57. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x

58. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x – 3x2

59. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x – 3x2 + 6x

60. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x – 3x2 + 6x + 6

61. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x – 3x2 + 6x + 6
= x3– 5x2 + 4x + 6
We will address the division operation of polynomials later-
after we understand more about the multiplication operation.

62. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example D. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x – 3x2 + 6x + 6
= x3– 5x2 + 4x + 6
We will address the division operation of polynomials later-
after we understand more about the multiplication operation.

63. We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example E. Multiply
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Polynomial Operations
(x – 3)(x2 – 2x – 2) = x3
x3

64. We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example E. Multiply
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Next collect all the x2 terms in the product,
Polynomial Operations
(x – 3)(x2 – 2x – 2) = x3

65. We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example E. Multiply
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Next collect all the x2 terms in the product, they are
–3*x2
Polynomial Operations
(x – 3)(x2 – 2x – 2) = x3
–3x2

66. We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example E. Multiply
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Next collect all the x2 terms in the product, they are
–3*x2 and x*(–2x)
Polynomial Operations
(x – 3)(x2 – 2x – 2) = x3
–3x2
–2x2

67. We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example E. Multiply
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Next collect all the x2 terms in the product, they are
–3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2.
Polynomial Operations
(x – 3)(x2 – 2x – 2) = x3 – 5x2
–3x2
–2x2

68. We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example E. Multiply
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Next collect all the x2 terms in the product, they are
–3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2.
Next collect the x–terms
Polynomial Operations
(x – 3)(x2 – 2x – 2) = x3 – 5x2

69. We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example E. Multiply
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Next collect all the x2 terms in the product, they are
–3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2.
Next collect the x–terms and they are x*(–2) + –3*(–2x) = 4x.
Polynomial Operations
(x – 3)(x2 – 2x – 2) = x3 – 5x2 + 4x
–2x
6x

70. We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example E. Multiply
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Next collect all the x2 terms in the product, they are
–3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2.
Next collect the x–terms and they are x*(–2) + –3*(–2x) = 4x.
The last term is the number term or –3(–2) = 6.
Polynomial Operations
(x – 3)(x2 – 2x – 2) = x3 – 5x2 + 4x + 6 (same as before)
6

71. Ex. A. Multiply the following monomials.
1. 3x2(–3x2)
11. 4x(3x – 5) – 9(6x – 7)
Polynomial Operations
2. –3x2(8x5) 3. –5x2(–3x3)
4. –12( )
6
–5x3
5. 24( x3)
8
–5
6. 6x2( )
3
2x3
7. –15x4( x5)
5
–2
C. Expand and simplify.
B. Fill in the degrees of the products.
8. #x(#x2 + # x + #) = #x? + #x? + #x?
9. #x2(#x4 + # x3 + #x2) = #x? + #x? + #x?
10. #x4(#x3 + # x2 + #x + #) = #x? + #x? + #x? + #x?
12. –x(2x + 7) + 3(4x – 2)
13. –3x(3x + 2) – 8x(7x – 5) 14. 5x(–5x + 9) + 6x(6x – 1)
15. 2x(–4x + 2) – 3x(2x – 1) – 3(4x – 2)
16. –4x(–7x + 9) – 2x(2x – 5) + 9(4x + 2)

72. 18. (x + 5)(x + 7)
Polynomial Operations
D. Expand and simplify. (Use any method.)
19. (x – 5)(x + 7)
20. (x + 5)(x – 7) 21. (x – 5)(x – 7)
22. (3x – 5)(2x + 4) 23. (–x + 5)(3x + 8)
24. (2x – 5)(2x + 5) 25. (3x + 7)(3x – 7)
26. (3x2 – 5)(x – 6) 27. (8x – 2)(–4x2 – 7)
28. (2x – 7)(x2 – 3x + 9) 29. (5x + 3)(2x2 – x + 5)
38. (x – 1)(x + 1) 39. (x – 1)(x2 + x + 1)
40. (x – 1)(x3 + x2 + x + 1)
41. (x – 1)(x4 + x3 + x2 + x + 1)
42. What do you think the answer is for
(x – 1)(x50 + x49 + …+ x2 + x + 1)?
30. (x – 1)(x – 1) 31. (x + 1)2
32. (2x – 3)2 33. (5x + 4)2
34. 2x(2x – 1)(3x + 2) 35. 4x(3x – 2)(2x + 3)
36. (x – 5)(2x – 1)(3x + 2) 37. (2x + 1)(3x + 1)(x – 2)