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 ๏ โthe sum of 2 and 3 times xโ
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions
5. Example A.
2 + 3x ๏ โthe sum of 2 and 3 times xโ
4x2 โ 5x
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions
6. Example A.
2 + 3x ๏ โthe sum of 2 and 3 times xโ
4x2 โ 5x ๏ โthe difference between 4 times the square of x
and 5 times xโ
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions
7. Example A.
2 + 3x ๏ โthe sum of 2 and 3 times xโ
4x2 โ 5x ๏ โthe difference between 4 times the square of x
and 5 times xโ
(3 โ 2x)2
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions
8. Example A.
2 + 3x ๏ โthe sum of 2 and 3 times xโ
4x2 โ 5x ๏ โthe difference between 4 times the square of x
and 5 times xโ
(3 โ 2x)2 ๏ โthe square of the difference of 3 and twice xโ
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions
9. Example A.
2 + 3x ๏ โthe sum of 2 and 3 times xโ
4x2 โ 5x ๏ โthe difference between 4 times the square of x
and 5 times xโ
(3 โ 2x)2 ๏ โthe square of the difference of 3 and twice xโ
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 ๏ โthe sum of 2 and 3 times xโ
4x2 โ 5x ๏ โthe difference between 4 times the square of x
and 5 times xโ
(3 โ 2x)2 ๏ โthe square of the difference of 3 and twice xโ
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, โ4x3, and 5x6 are monomials.
Polynomial Expressions
11. Example A.
2 + 3x ๏ โthe sum of 2 and 3 times xโ
4x2 โ 5x ๏ โthe difference between 4 times the square of x
and 5 times xโ
(3 โ 2x)2 ๏ โthe square of the difference of 3 and twice xโ
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Example B. Evaluate the monomials if y = โ4
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, โ4x3, and 5x6 are monomials.
Polynomial Expressions
12. Example A.
2 + 3x ๏ โthe sum of 2 and 3 times xโ
4x2 โ 5x ๏ โthe difference between 4 times the square of x
and 5 times xโ
(3 โ 2x)2 ๏ โthe square of the difference of 3 and twice xโ
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Example B. Evaluate the monomials if y = โ4
a. 3y2
3y2 ๏ 3(โ4)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, โ4x3, and 5x6 are monomials.
Polynomial Expressions
13. Example A.
2 + 3x ๏ โthe sum of 2 and 3 times xโ
4x2 โ 5x ๏ โthe difference between 4 times the square of x
and 5 times xโ
(3 โ 2x)2 ๏ โthe square of the difference of 3 and twice xโ
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Example B. Evaluate the monomials if y = โ4
a. 3y2
3y2 ๏ 3(โ4)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, โ4x3, and 5x6 are monomials.
Polynomial Expressions
22. b. โ3y2 (y = โ4)
โ3y2 ๏ โ3(โ4)2
= โ3(16) = โ48.
c. โ3y3
โ3y3 ๏ โ 3(โ4)3
= โ 3(โ64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ยฑ #xN-1 ยฑ โฆ ยฑ #x1 ยฑ #
where # can be any number.
Polynomial Expressions
Polynomial Expressions
23. b. โ3y2 (y = โ4)
โ3y2 ๏ โ3(โ4)2
= โ3(16) = โ48.
c. โ3y3
โ3y3 ๏ โ 3(โ4)3
= โ 3(โ64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ยฑ #xN-1 ยฑ โฆ ยฑ #x1 ยฑ #
where # can be any number.
For example, 4x + 7,
Polynomial Expressions
Polynomial Expressions
24. b. โ3y2 (y = โ4)
โ3y2 ๏ โ3(โ4)2
= โ3(16) = โ48.
c. โ3y3
โ3y3 ๏ โ 3(โ4)3
= โ 3(โ64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ยฑ #xN-1 ยฑ โฆ ยฑ #x1 ยฑ #
where # can be any number.
For example, 4x + 7, โ3x2 โ 4x + 7,
Polynomial Expressions
Polynomial Expressions
25. b. โ3y2 (y = โ4)
โ3y2 ๏ โ3(โ4)2
= โ3(16) = โ48.
c. โ3y3
โ3y3 ๏ โ 3(โ4)3
= โ 3(โ64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ยฑ #xN-1 ยฑ โฆ ยฑ #x1 ยฑ #
where # can be any number.
For example, 4x + 7, โ3x2 โ 4x + 7, โ5x4 + 1 are polynomials,
Polynomial Expressions
Polynomial Expressions
26. b. โ3y2 (y = โ4)
โ3y2 ๏ โ3(โ4)2
= โ3(16) = โ48.
c. โ3y3
โ3y3 ๏ โ 3(โ4)3
= โ 3(โ64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ยฑ #xN-1 ยฑ โฆ ยฑ #x1 ยฑ #
where # can be any number.
For example, 4x + 7, โ3x2 โ 4x + 7, โ5x4 + 1 are polynomials,
x
1
is not a polynomial.whereas the expression
Polynomial Expressions
Polynomial Expressions
27. Example C. Evaluate the polynomial 4x2 โ 3x3 if x = โ3.
Polynomial Expressions
28. Example C. Evaluate the polynomial 4x2 โ 3x3 if x = โ3.
The polynomial 4x2 โ 3x3 is the combination of two
monomials; 4x2 and โ3x3.
Polynomial Expressions
29. Example C. Evaluate the polynomial 4x2 โ 3x3 if x = โ3.
The polynomial 4x2 โ 3x3 is the combination of two
monomials; 4x2 and โ3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Polynomial Expressions
30. Example C. Evaluate the polynomial 4x2 โ 3x3 if x = โ3.
The polynomial 4x2 โ 3x3 is the combination of two
monomials; 4x2 and โ3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (โ3) in the expression,
Polynomial Expressions
31. Example C. Evaluate the polynomial 4x2 โ 3x3 if x = โ3.
The polynomial 4x2 โ 3x3 is the combination of two
monomials; 4x2 and โ3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (โ3) in the expression, we get
4(โ3)2 โ 3(โ3)3
Polynomial Expressions
32. Example C. Evaluate the polynomial 4x2 โ 3x3 if x = โ3.
The polynomial 4x2 โ 3x3 is the combination of two
monomials; 4x2 and โ3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (โ3) in the expression, we get
4(โ3)2 โ 3(โ3)3
= 4(9) โ 3(โ27)
Polynomial Expressions
33. Example C. Evaluate the polynomial 4x2 โ 3x3 if x = โ3.
The polynomial 4x2 โ 3x3 is the combination of two
monomials; 4x2 and โ3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (โ3) in the expression, we get
4(โ3)2 โ 3(โ3)3
= 4(9) โ 3(โ27)
= 36 + 81
= 117
Polynomial Expressions
34. Example C. Evaluate the polynomial 4x2 โ 3x3 if x = โ3.
The polynomial 4x2 โ 3x3 is the combination of two
monomials; 4x2 and โ3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (โ3) in the expression, we get
4(โ3)2 โ 3(โ3)3
= 4(9) โ 3(โ27)
= 36 + 81
= 117
Given a polynomial, each monomial is called a term.
Polynomial Expressions
35. Example C. Evaluate the polynomial 4x2 โ 3x3 if x = โ3.
The polynomial 4x2 โ 3x3 is the combination of two
monomials; 4x2 and โ3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (โ3) in the expression, we get
4(โ3)2 โ 3(โ3)3
= 4(9) โ 3(โ27)
= 36 + 81
= 117
Given a polynomial, each monomial is called a term.
#xN ยฑ #xN-1 ยฑ โฆ ยฑ #x ยฑ #
terms
Polynomial Expressions
36. Example C. Evaluate the polynomial 4x2 โ 3x3 if x = โ3.
The polynomial 4x2 โ 3x3 is the combination of two
monomials; 4x2 and โ3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (โ3) in the expression, we get
4(โ3)2 โ 3(โ3)3
= 4(9) โ 3(โ27)
= 36 + 81
= 117
Given a polynomial, each monomial is called a term.
#xN ยฑ #xN-1 ยฑ โฆ ยฑ #x ยฑ #
terms
Therefore the polynomial โ3x2 โ 4x + 7 has 3 terms,
โ3x2 , โ4x 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 โ3x2 โ 4x + 7 is โ3x2,
Polynomial Expressions
39. Each term is addressed by the variable part. Hence the
x2-term of the โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
Polynomial Expressions
40. Each term is addressed by the variable part. Hence the
x2-term of the โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 is โ3 .
Polynomial Expressions
43. Each term is addressed by the variable part. Hence the
x2-term of the โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 is โ3 .
Operations with Polynomials
Polynomial Expressions
44. Each term is addressed by the variable part. Hence the
x2-term of the โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 is โ3 .
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 โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 is โ3 .
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 โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 is โ3 .
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 โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 is โ3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 โ 5x2 = โ2x2.
Operations with Polynomials
Polynomial Expressions
48. Each term is addressed by the variable part. Hence the
x2-term of the โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 is โ3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 โ 5x2 = โ2x2.
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 โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 is โ3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 โ 5x2 = โ2x2.
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 โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 is โ3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 โ 5x2 = โ2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , โ1xN as โxN.
Operations with Polynomials
Polynomial Expressions
51. Each term is addressed by the variable part. Hence the
x2-term of the โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 is โ3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 โ 5x2 = โ2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , โ1xN as โxN.
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 โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 is โ3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 โ 5x2 = โ2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , โ1xN as โxN.
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 โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 is โ3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 โ 5x2 = โ2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , โ1xN as โxN.
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 โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 is โ3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 โ 5x2 = โ2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , โ1xN as โxN.
When multiplying a number with a term, we multiply it with the
coefficient. Hence, 3(5x) = (3*5)x =15x,
and โ2(โ4x) = (โ2)(โ4)x = 8x.
Operations with Polynomials
Polynomial Expressions
55. Each term is addressed by the variable part. Hence the
x2-term of the โ3x2 โ 4x + 7 is โ3x2, the x-term is โ4x,
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 โ3x2 is โ3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 โ 5x2 = โ2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , โ1xN as โxN.
When multiplying a number with a term, we multiply it with the
coefficient. Hence, 3(5x) = (3*5)x =15x,
and โ2(โ4x) = (โ2)(โ4)x = 8x.
Operations with Polynomials
When multiplying a number with a polynomial, we may
expand using the distributive law: A(B ยฑ C) = AB ยฑ AC.
Polynomial Expressions
65. Example D. Expand and simplify.
a. 3(2x โ 4) + 2(4 โ 5x)
= 6x โ 12 + 8 โ 10x
= โ4x โ 4
b. โ3(x2 โ 3x + 5) โ 2(โx2 โ 4x โ 6)
= โ3x2 + 9x โ 15 + 2x2 + 8x +12
= โx2 + 17x โ 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 โ 4) + 2(4 โ 5x)
= 6x โ 12 + 8 โ 10x
= โ4x โ 4
b. โ3(x2 โ 3x + 5) โ 2(โx2 โ 4x โ 6)
= โ3x2 + 9x โ 15 + 2x2 + 8x +12
= โx2 + 17x โ 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(โ4x) =
c. 3x2(2x3 โ 4x)
=
67. Example D. Expand and simplify.
a. 3(2x โ 4) + 2(4 โ 5x)
= 6x โ 12 + 8 โ 10x
= โ4x โ 4
b. โ3(x2 โ 3x + 5) โ 2(โx2 โ 4x โ 6)
= โ3x2 + 9x โ 15 + 2x2 + 8x +12
= โx2 + 17x โ 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(โ4x) =
c. 3x2(2x3 โ 4x)
=
68. Example D. Expand and simplify.
a. 3(2x โ 4) + 2(4 โ 5x)
= 6x โ 12 + 8 โ 10x
= โ4x โ 4
b. โ3(x2 โ 3x + 5) โ 2(โx2 โ 4x โ 6)
= โ3x2 + 9x โ 15 + 2x2 + 8x +12
= โx2 + 17x โ 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(โ4x) =
c. 3x2(2x3 โ 4x)
=
69. Example D. Expand and simplify.
a. 3(2x โ 4) + 2(4 โ 5x)
= 6x โ 12 + 8 โ 10x
= โ4x โ 4
b. โ3(x2 โ 3x + 5) โ 2(โx2 โ 4x โ 6)
= โ3x2 + 9x โ 15 + 2x2 + 8x +12
= โx2 + 17x โ 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(โ4x) = 3(โ4)x2x = โ12x3
c. 3x2(2x3 โ 4x)
=
70. Example D. Expand and simplify.
a. 3(2x โ 4) + 2(4 โ 5x)
= 6x โ 12 + 8 โ 10x
= โ4x โ 4
b. โ3(x2 โ 3x + 5) โ 2(โx2 โ 4x โ 6)
= โ3x2 + 9x โ 15 + 2x2 + 8x +12
= โx2 + 17x โ 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(โ4x) = 3(โ4)x2x = โ12x3
c. 3x2(2x3 โ 4x) distribute
= 6x5 โ 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 โ 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 โ 1) + 2(2x โ 1)
a. (3x + 2)(2x โ 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 โ 1) + 2(2x โ 1)
= 6x2 โ 3x + 4x โ 2
a. (3x + 2)(2x โ 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 โ 1) + 2(2x โ 1)
= 6x2 โ 3x + 4x โ 2
= 6x2 + x โ 2
a. (3x + 2)(2x โ 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 โ 1)(2x2 + 3x โ4)
= 3x(2x โ 1) + 2(2x โ 1)
= 6x2 โ 3x + 4x โ 2
= 6x2 + x โ 2
a. (3x + 2)(2x โ 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 โ 1)(2x2 + 3x โ4)
= 3x(2x โ 1) + 2(2x โ 1)
= 6x2 โ 3x + 4x โ 2
= 6x2 + x โ 2
= 2x(2x2 + 3x โ4) โ1(2x2 + 3x โ 4)
a. (3x + 2)(2x โ 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 โ 1)(2x2 + 3x โ4)
= 3x(2x โ 1) + 2(2x โ 1)
= 6x2 โ 3x + 4x โ 2
= 6x2 + x โ 2
= 2x(2x2 + 3x โ4) โ1(2x2 + 3x โ 4)
= 4x3 + 6x2 โ 8x โ 2x2 โ 3x + 4
a. (3x + 2)(2x โ 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 โ 1)(2x2 + 3x โ4)
= 3x(2x โ 1) + 2(2x โ 1)
= 6x2 โ 3x + 4x โ 2
= 6x2 + x โ 2
= 2x(2x2 + 3x โ4) โ1(2x2 + 3x โ 4)
= 4x3 + 6x2 โ 8x โ 2x2 โ 3x + 4
= 4x3 + 4x2 โ 11x + 4
a. (3x + 2)(2x โ 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 โ 1)(2x2 + 3x โ4)
= 3x(2x โ 1) + 2(2x โ 1)
= 6x2 โ 3x + 4x โ 2
= 6x2 + x โ 2
= 2x(2x2 + 3x โ4) โ1(2x2 + 3x โ 4)
= 4x3 + 6x2 โ 8x โ 2x2 โ 3x + 4
= 4x3 + 4x2 โ 11x + 4
a. (3x + 2)(2x โ 1)
Note that if we did (2x โ 1)(3x + 2) or (2x2 + 3x โ4)(2x โ 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 โ 1)(2x2 + 3x โ4)
= 3x(2x โ 1) + 2(2x โ 1)
= 6x2 โ 3x + 4x โ 2
= 6x2 + x โ 2
= 2x(2x2 + 3x โ4) โ1(2x2 + 3x โ 4)
= 4x3 + 6x2 โ 8x โ 2x2 โ 3x + 4
= 4x3 + 4x2 โ 11x + 4
a. (3x + 2)(2x โ 1)
Note that if we did (2x โ 1)(3x + 2) or (2x2 + 3x โ4)(2x โ 1)
instead, we get the same answers. (Check this.)
Fact. If P and Q are two polynomials then PQ โก 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 โ 1)(2x2 + 3x โ4)
= 3x(2x โ 1) + 2(2x โ 1)
= 6x2 โ 3x + 4x โ 2
= 6x2 + x โ 2
= 2x(2x2 + 3x โ4) โ1(2x2 + 3x โ 4)
= 4x3 + 6x2 โ 8x โ 2x2 โ 3x + 4
= 4x3 + 4x2 โ 11x + 4
a. (3x + 2)(2x โ 1)
Note that if we did (2x โ 1)(3x + 2) or (2x2 + 3x โ4)(2x โ 1)
instead, we get the same answers. (Check this.)
Fact. If P and Q are two polynomials then PQ โก QP.
A shorter way to multiply is to bypass the 2nd step and use the
general distributive law.
86. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..
Polynomial Operations
87. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Polynomial Operations
88. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Example G. Expand
a. (x + 3)(x โ 4)
Polynomial Operations
89. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Example G. Expand
a. (x + 3)(x โ 4)
= x2
Polynomial Operations
90. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Example G. Expand
a. (x + 3)(x โ 4)
= x2 โ 4x
Polynomial Operations
91. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Example G. Expand
a. (x + 3)(x โ 4)
= x2 โ 4x + 3x
Polynomial Operations
92. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Example G. Expand
a. (x + 3)(x โ 4)
= x2 โ 4x + 3x โ 12
Polynomial Operations
93. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Example G. Expand
a. (x + 3)(x โ 4)
= x2 โ 4x + 3x โ 12 simplify
= x2 โ x โ 12
Polynomial Operations
94. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Example G. Expand
a. (x + 3)(x โ 4)
= x2 โ 4x + 3x โ 12 simplify
= x2 โ x โ 12
b. (x โ 3)(x2 โ 2x โ 2)
Polynomial Operations
95. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Example G. Expand
a. (x + 3)(x โ 4)
= x2 โ 4x + 3x โ 12 simplify
= x2 โ x โ 12
b. (x โ 3)(x2 โ 2x โ 2)
Polynomial Operations
= x3
96. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Example G. Expand
a. (x + 3)(x โ 4)
= x2 โ 4x + 3x โ 12 simplify
= x2 โ x โ 12
b. (x โ 3)(x2 โ 2x โ 2)
Polynomial Operations
= x3 โ 2x2
97. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Example G. Expand
a. (x + 3)(x โ 4)
= x2 โ 4x + 3x โ 12 simplify
= x2 โ x โ 12
b. (x โ 3)(x2 โ 2x โ 2)
Polynomial Operations
= x3 โ 2x2 โ 2x
98. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Example G. Expand
a. (x + 3)(x โ 4)
= x2 โ 4x + 3x โ 12 simplify
= x2 โ x โ 12
b. (x โ 3)(x2 โ 2x โ 2)
Polynomial Operations
= x3 โ 2x2 โ 2x โ 3x2
99. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Example G. Expand
a. (x + 3)(x โ 4)
= x2 โ 4x + 3x โ 12 simplify
= x2 โ x โ 12
b. (x โ 3)(x2 โ 2x โ 2)
Polynomial Operations
= x3 โ 2x2 โ 2x โ 3x2 + 6x
100. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Example G. Expand
a. (x + 3)(x โ 4)
= x2 โ 4x + 3x โ 12 simplify
= x2 โ x โ 12
b. (x โ 3)(x2 โ 2x โ 2)
Polynomial Operations
= x3 โ 2x2 โ 2x โ 3x2 + 6x + 6
101. General Distributive Rule:
(A ยฑ B ยฑ C ยฑ ..)(a ยฑ b ยฑ c ..)
= Aa ยฑ Ab ยฑ Ac ..ยฑ Ba ยฑ Bb ยฑ Bc ..ยฑCa ยฑ Cb ยฑ Cc ..
Example G. Expand
a. (x + 3)(x โ 4)
= x2 โ 4x + 3x โ 12 simplify
= x2 โ x โ 12
b. (x โ 3)(x2 โ 2x โ 2)
Polynomial Operations
= x3 โ 2x2 โ 2x โ 3x2 + 6x + 6
= x3โ 5x2 + 4x + 6
We will address the division operation of polynomials later-
after we understand more about the multiplication operation.