Subtract 8x
–
5
from
–
5x + 3x
2
Remember
The subtraction operation is neither
commutative
nor
associative
. The order of the subtracted terms and the order
in which the operations are performed is
important
.
It is not
commutative
because 5 -
3 ≠ 3
- 5
It is not
associative
because (3 - 5) -
7 ≠ 3
- (5 - 7)
34

4.4: Addition and Subtraction of Polynomials
Examples: Using the Vertical Method
1.
Add 3x
–
5
and x
2
+
2x
+ 4
2.
Add 2x
2
–
5x, 3x
2
+ 2 and 6x
–
3
3.
Subtract 5x
–
3
from 8x
–
7
4.
Subtract 5x
2
–
3
x + 4
from 8x
2
+ 5x
–
3
35

4.5: Multiplication of Polynomials and Special Products
Objectives:
4.5.1: Find the product of a monomial and a
polynomial
4.5.2: Find the product of two binomials
4.5.3: Square a binomial
4.5.4: Find the product of two binomials that differ
only in sign
36

4.5: To Multiply Monomials
Example:
1.
Multiply 3x
2
y and 2x
3
y
5
Remember:
The multiplication operation is
commutative
and
associative
.
The order of the multiplied terms and the order in which the
operations are performed is
not important
.
It is
commutative
because (2)(3) = (3)(2) = 6
It is
associative
because ((2) (3))(4) = (2)((3)(4)) = 24
37

4.5.1: Find the Product of a Monomial and a Polynomial
Example:
1.
Multiply 2x + 3 by x
2.
Multiply 2a
3
+ 4a by 3a
2
3.
3x(4x
3
+ x
2
+ 2)
4.
–
5
c(4c
2
–
8c)
5.
3c
2
d
2
(7cd
2
–
5c
2
d
3
)
6.
3x
2
y and 2x
3
y
5
38

4.5.2: Find the Product of Two Binomials
Example:
1.
Multiply x + 2 by x + 3
2.
Multiply a
–
3 by a
–
4
39

4.5.2: Find the Product of Two Binomials Using FOIL
Method
Example:
Multiply (x + 2) by (x + 3)
40

4.5.2: Find the Product of Two Binomials Using FOIL
Method
Examples:
1.
Multiply (x + 4) by (x + 5)
2.
Multiply (x
–
7) by (x + 3)
3.
Multiply (4x
–
3) by (3x + 2)
4.
Multiply (3x
–
5y) by (2x
–
5y)
41

4.5.2: Find the Product of Two Binomials
Examples:
Using the Vertical Method for Polynomials with
Three or More Terms
1.
Multiply (x
2
–
5x + 8) by (x + 3)
42

4.5.3: Square a Binomial
Squaring
a binomial always results in
three
terms.
Examples:
(x + y)
2
= (x + y)(x + y) = x
2
+ xy + xy + y
2
= x
2
+ 2xy + y
2
(x
–
y)
2
= (x
–
y)(x
–
y) = x
2
–
xy
–
xy + y
2
= x
2
–
2xy + y
2
To Square a Binomial
43

4.5.3: Square a Binomial
Examples:
1.
(x + 3)
2
2.
(3a + 4b)
2
3.
(y
–
5)
2
4.
(5c
–
3d)
2
Note: (y + 4)
2
≠
y
2
+ 4
2
or y
2
+ 16
To Square a Binomial
44

4.5.4: Find the Product of Two Binomials that Differ
Only in Sign
Example:
(x + y)(x
–
y) = x
2
–
xy + xy
–
y
2
= x
2
–
y
2
Examples:
1.

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- Fall '18
- jane
- Accounting, Algebra, Addition, Exponents, Exponentiation