polynomial expressions and operations

1. Polynomial Expressions

2. A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions

3. Example A.
2 + 3x 
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions

4. Example A.
2 + 3x  “the sum of 2 and 3 times x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions

5. Example A.
2 + 3x  “the sum of 2 and 3 times x”
4x2 – 5x
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions

6. Example A.
2 + 3x  “the sum of 2 and 3 times x”
4x2 – 5x  “the difference between 4 times the square of x
and 5 times x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions

7. Example A.
2 + 3x  “the sum of 2 and 3 times x”
4x2 – 5x  “the difference between 4 times the square of x
and 5 times x”
(3 – 2x)2
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions

8. Example A.
2 + 3x  “the sum of 2 and 3 times x”
4x2 – 5x  “the difference between 4 times the square of x
and 5 times x”
(3 – 2x)2  “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions

9. Example A.
2 + 3x  “the sum of 2 and 3 times x”
4x2 – 5x  “the difference between 4 times the square of x
and 5 times x”
(3 – 2x)2  “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
An expression of the form #xN, where the exponent N is a
non-negative integer and # is a number, is called a monomial
(one-term).
Polynomial Expressions

10. Example A.
2 + 3x  “the sum of 2 and 3 times x”
4x2 – 5x  “the difference between 4 times the square of x
and 5 times x”
(3 – 2x)2  “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
An expression of the form #xN, where the exponent N is a
non-negative integer and # is a number, is called a monomial
(one-term).
For example, 3x2, –4x3, and 5x6 are monomials.
Polynomial Expressions

11. Example A.
2 + 3x  “the sum of 2 and 3 times x”
4x2 – 5x  “the difference between 4 times the square of x
and 5 times x”
(3 – 2x)2  “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Example B. Evaluate the monomials if y = –4
a. 3y2
An expression of the form #xN, where the exponent N is a
non-negative integer and # is a number, is called a monomial
(one-term).
For example, 3x2, –4x3, and 5x6 are monomials.
Polynomial Expressions

12. Example A.
2 + 3x  “the sum of 2 and 3 times x”
4x2 – 5x  “the difference between 4 times the square of x
and 5 times x”
(3 – 2x)2  “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Example B. Evaluate the monomials if y = –4
a. 3y2
3y2  3(–4)2
An expression of the form #xN, where the exponent N is a
non-negative integer and # is a number, is called a monomial
(one-term).
For example, 3x2, –4x3, and 5x6 are monomials.
Polynomial Expressions

13. Example A.
2 + 3x  “the sum of 2 and 3 times x”
4x2 – 5x  “the difference between 4 times the square of x
and 5 times x”
(3 – 2x)2  “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Example B. Evaluate the monomials if y = –4
a. 3y2
3y2  3(–4)2
= 3(16) = 48
An expression of the form #xN, where the exponent N is a
non-negative integer and # is a number, is called a monomial
(one-term).
For example, 3x2, –4x3, and 5x6 are monomials.
Polynomial Expressions

14. b. –3y2 (y = –4)
Polynomial Expressions

15. b. –3y2 (y = –4)
–3y2  –3(–4)2
Polynomial Expressions

16. b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
Polynomial Expressions

17. b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
Polynomial Expressions

18. b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
Polynomial Expressions

19. b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64)
Polynomial Expressions

20. b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
Polynomial Expressions

21. b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
Polynomial Expressions
Polynomial Expressions

22. b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ± #xN-1 ± … ± #x1 ± #
where # can be any number.
Polynomial Expressions
Polynomial Expressions

23. b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ± #xN-1 ± … ± #x1 ± #
where # can be any number.
For example, 4x + 7,
Polynomial Expressions
Polynomial Expressions

24. b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ± #xN-1 ± … ± #x1 ± #
where # can be any number.
For example, 4x + 7, –3x2 – 4x + 7,
Polynomial Expressions
Polynomial Expressions

25. b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ± #xN-1 ± … ± #x1 ± #
where # can be any number.
For example, 4x + 7, –3x2 – 4x + 7, –5x4 + 1 are polynomials,
Polynomial Expressions
Polynomial Expressions

26. b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ± #xN-1 ± … ± #x1 ± #
where # can be any number.
For example, 4x + 7, –3x2 – 4x + 7, –5x4 + 1 are polynomials,
x
1
is not a polynomial.whereas the expression
Polynomial Expressions
Polynomial Expressions

27. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
Polynomial Expressions

28. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3.
Polynomial Expressions

29. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Polynomial Expressions

30. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression,
Polynomial Expressions

31. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
Polynomial Expressions

32. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
Polynomial Expressions

33. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
= 36 + 81
= 117
Polynomial Expressions

34. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
= 36 + 81
= 117
Given a polynomial, each monomial is called a term.
Polynomial Expressions

35. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
= 36 + 81
= 117
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Polynomial Expressions

36. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
= 36 + 81
= 117
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Therefore the polynomial –3x2 – 4x + 7 has 3 terms,
–3x2 , –4x and + 7.
Polynomial Expressions

37. Each term is addressed by the variable part.
Polynomial Expressions

38. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2,
Polynomial Expressions

39. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
Polynomial Expressions

40. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
Polynomial Expressions

41. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term.
Polynomial Expressions

42. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Polynomial Expressions

43. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Operations with Polynomials
Polynomial Expressions

44. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Operations with Polynomials
Polynomial Expressions

45. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
Operations with Polynomials
Polynomial Expressions

46. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x
Operations with Polynomials
Polynomial Expressions

47. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Operations with Polynomials
Polynomial Expressions

48. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined.
Operations with Polynomials
Polynomial Expressions

49. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Operations with Polynomials
Polynomial Expressions

50. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
Operations with Polynomials
Polynomial Expressions

51. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient.
Operations with Polynomials
Polynomial Expressions

52. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient. Hence, 3(5x) = (3*5)x
Operations with Polynomials
Polynomial Expressions

53. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient. Hence, 3(5x) = (3*5)x =15x,
Operations with Polynomials
Polynomial Expressions

54. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
Operations with Polynomials
Polynomial Expressions

55. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
Operations with Polynomials
When multiplying a number with a polynomial, we may
expand using the distributive law: A(B ± C) = AB ± AC.
Polynomial Expressions

56. Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
Polynomial Expressions

57. Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
Polynomial Expressions

58. Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
Polynomial Expressions

59. Example D. 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 Expressions

60. Example D. 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 Expressions

61. Example D. 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 Expressions

62. Example D. 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.

63. Example D. 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 E.
a. (3x2)(2x3) =
b. 3x2(–4x) =
c. 3x2(2x3 – 4x)
=

64. Example D. 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 E.
a. (3x2)(2x3) = 3*2x2x3
b. 3x2(–4x) =
c. 3x2(2x3 – 4x)
=

65. Example D. 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 E.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) =
c. 3x2(2x3 – 4x)
=

66. Example D. 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 E.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) = 3(–4)x2x = –12x3
c. 3x2(2x3 – 4x)
=

67. Example D. 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 E.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) = 3(–4)x2x = –12x3
c. 3x2(2x3 – 4x) distribute
= 6x5 – 12x3

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

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

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

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

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

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

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

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

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

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

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

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

80. General Distributive Rule:
Polynomial Operations

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

100. Polynomials in two or more variables.
Polynomial Expressions

101. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Polynomial Expressions

102. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined.
Polynomial Expressions

103. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2

104. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2

105. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2

106. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2

107. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.

108. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y. If only one number is given, the result is a formula.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.

109. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y. If only one number is given, the result is a formula.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2

110. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y. If only one number is given, the result is a formula.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
= 16y – 32y – 6y2 = –16y – 6y2

111. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y. If only one number is given, the result is a formula.
If both numbers are given, then we get a numerical output.
We may do this for x, y and z or even more variables.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
= 16y – 32y – 6y2 = –16y – 6y2

112. Ex. A. Evaluate each monomials with the given values.
3. 2x2 with x = 1 and x = –1 4. –2x2 with x = 1 and x = –1
5. 5y3 with y = 2 and y = –2 6. –5y3 with y = 2 and y = –2
1. 2x with x = 1 and x = –1 2. –2x with x = 1 and x = –1
7. 5z4 with z = 2 and z = –2 8. –5y4 with z = 2 and z = –2
B. Evaluate each monomials with the given values.
9. 2x2 – 3x + 2 with x = 1 and x = –1
10. –2x2 + 4x – 1 with x = 2 and x = –2
11. 3x2 – x – 2 with x = 3 and x = –3
12. –3x2 – x + 2 with x = 3 and x = –3
13. –2x3 – x2 + 4 with x = 2 and x = –2
14. –2x3 – 5x2 – 5 with x = 3 and x = –3
C. Expand and simplify.
15. 5(2x – 4) + 3(4 – 5x) 16. 5(2x – 4) – 3(4 – 5x)
17. –2(3x – 8) + 3(4 – 9x) 18. –2(3x – 8) – 3(4 – 9x)
19. 7(–2x – 7) – 3(4 – 3x) 20. –5(–2 – 8x) + 7(–2 – 11x)
Polynomial Expressions

113. 21. x2 – 3x + 5 + 2(–x2 – 4x – 6)
22. x2 – 3x + 5 – 2(–x2 – 4x – 6)
23. 2(x2 – 3x + 5) + 5(–x2 – 4x – 6)
24. 2(x2 – 3x + 5) – 5(–x2 – 4x – 6)
25. –2(3x2 – 2x + 5) + 5(–4x2 – 4x – 3)
26. –2(3x2 – 2x + 5) – 5(–4x2 – 4x – 3)
27. 4(3x3 – 5x2) – 9(6x2 – 7x) – 5(– 8x – 2)
29. Simplify 2(3xy – xy2) – 2(2xy – xy2) then evaluated it
with x = –1, afterwards evaluate it at (–1, 2) for (x, y)
30. Simplify x2 – 2(3xy – x2) – 2(y2 – xy) then evaluated it
with y = –2, afterwards evaluate it at (–1, –2) for (x, y)
31. Simplify x2 – 2(3xy – z2) – 2(z2 – x2) then evaluated it
with x = –1, y = – 2 and z = 3.
Polynomial Expressions
28. –6(7x2 + 5x – 9) – 7(–3x2 – 2x – 7)