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
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
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.
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) =
b. 3x2(–4x) =
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
b. 3x2(–4x) =
c. 3x2(2x3 – 4x)
=
68. 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)
=
69. 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)
=
70. 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
71. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
72. 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)
73. 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)
74. 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)
75. 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)
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
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)
a. (3x + 2)(2x – 1)
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
a. (3x + 2)(2x – 1)
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)
80. 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.)
81. 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.
82. 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.
87. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
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)
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
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
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
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
Polynomial Operations
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
Polynomial Operations
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
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
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
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
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
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
100. 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
101. 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.