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1.2 - ProductsCommutative Properties
Addition: a b b a Multiplication: a b b a
m r
12t
5 y
8 z
5y
8z
12 t
r m
Associative Properties
Addition: a b c a b c Multiplication: a b c a b c
92mr 17q r
5 3 6
2 7 3 5 3 6
2 7 3
17q r
92m r
1.2 - Products
Distributive Property of Multiplication
a b c ab ac
4 7k 4 6 2x y z
5 x y
3 2 7x 5 5x y6 21x
4 24 8x y z 4 7k
a b c ab ac
3k
1.2 - Products
Product Rule for Exponents
If m and n are positive integers and a is a real number, then
Examples:3 54 4
m n m na a a
3 54 84
3 27 7 3 27 57
8 69 9 149 3 7 93 3 3 193
6 2 3s s s 11s 7m m 8m
1.2 - Products
Power Rule for Exponents
If m and n are positive integers and a is a real number, then
Examples:
423
nm m na a
2 2 2 23 3 3 3
2 43
83
1049 4 109 409
36z 6 3z 8 2y
83
18z 28y 16y
1.2 - Products
Power of a Product Rule
If m, n, and r are positive integers and a and b are real numbers, then
Examples:
424y
rm n m r n ra b a b
33x
1 4 2 44 y
327x
34 22p q r
4 84 y
3 33 x
1 3 4 3 2 3 1 32 p q r
3 12 6 32 p q r 12 6 38p q r
8256y
1.2 - Products
Multiplying Monomials by Monomials
Examples:
10 9x x
3 78 11x x
45x x
290x
1088x
55x
1.2 - Products
Multiplying Monomials by Polynomials
Examples:
3 25 3 2x x x
24 4 3x x x
48 7 1x x
34x 216x 12x
556x 8x
515x 45x 310x
1.2 - Products
Multiplying Two Binomials using FOIL
6 2 3 3x x
2 1 5 6x x
First terms Last termsInner termsOuter terms
210x 12x 5x 6
210x 7x 6
218x 18x 6x 6
218x 24x 6
1.2 - Products
Multiplying Two Binomials using FOIL
22 3 4y y
First terms Last termsInner termsOuter terms
22 4y
32y 28y 3y 12
32y 28y 3y 12
2 24 4y y
4y 24y 16 24y 4y 28y 16
1.2 - Products
Squaring Binomials
24 5x 216x
216 40 25x x
2 2 2
2 2 2
2
2
a b a ab b
a b a ab b
23 6x 29x
29 36 36x x
2 20x 25
2 18x 36
1.2 - Products
Multiplying Two Polynomials
Examples:
25 10 3x x x
24 5 3 4x x x
3x 210x 3x 25x 50x 15
3x 215x 47x 15
312x 216x 23x 4x 15x 20 312x 213x 11x 20
1.2 - Products
72 isxifx
Evaluate the following:
27 5
483 yandxifxy
384 54
20
1.3 – Sums and DifferencesAlgebraic Expression - A combination of operations on variables and numbers.
1225 3 xandzifxz
Evaluate the following:
121225 3 xandzif
1825
117 18
1.3 – Sums and Differences
5 3 33 7 2 1 3 2x x x x x
Simplify each polynomial.
5 3 33 7 2 1 3 2x x x x x
53x 34x 1
1.3 – Sums and Differences
3 2 3 24 10 1 4 11x x x x
Simplify each polynomial.
3 2 3 24 10 1 4 11x x x x
38x 211x 12
1.3 – Sums and Differences
A Number as a Product of Prime Numbers
72
36
9
2
182
33222 23 32
2
Factor Trees
210
105
7
2
35
5
7532
3
3 3
1.4 - Factorizations
A Number as a Product of Prime Numbers
Factor Trees
1.4 - Factorizations
A Number as a Product of Prime Numbers
1.4 - Factorizations
Factoring by Grouping 4x2 2x 10x 5+++
x2 12 x 5 12 x
12 x 52 x
1.4 - Factorizations
2x2 3x 8x 12+--
x 32 x 4 32 x
32 x 4x
Factoring by Grouping 1.4 - Factorizations
35x2 10x 14x 4-+-
x5 27 x 2 27 x
27 x 25 x
Factoring by Grouping 1.4 - Factorizations
Factoring Trinomials 2x bx c 2 27 50x x
Factors of 50 are:
x x
2 25x x
1, 50 2, 25 5,10
1.4 - Factorizations
Factoring Trinomials 2x bx c 2 5 36x x
Factors of 36 are:
x x
4 9x x
1, 36 2, 18 3, 12 4, 9 6, 6
1.4 - Factorizations
Factoring Trinomials 2x bx c
Factors of 9 are:
4 x x
4 3 3x x
24 24 36x x
24 6 9x x
1, 9 3, 3
1.4 - Factorizations
Factoring Trinomials 2ax bx c 22 11 12x x
Product of 2 and 12:
Factors of 24 are:
24
1, 24 2, 12 3, 8
3, 8Factors of 24 that combine to 11:
2x2 3x 8x 12+--
x 32 x 4 32 x
32 x 4x
4, 6
1.4 - Factorizations
Factoring Trinomials 2ax bx c 2 212 16 3a ab b
Product of 12 and 3:
Factors of 36 are:
36
1, 36 2, 18 3, 12
2, 18Factors of 36 that combine to 16:
12a2 2ab 18ab 3b2--+
a2 ba6 b3 ba6
ba6 ba 32
4, 9 6, 6
1.4 - Factorizations
3 1 3 1s s 29 1s
2 81p
24 100x
9 9p p
Not the difference
The Difference of Two Squares
1.4 - Factorizations
3 38 8
5 5c c
2 9
6425
c
2 2121 49x y 11 7 11 7x y x y
The Difference of Two Squares
1.4 - Factorizations
The Sum and Difference of Two Cubes
1.4 - Factorizations
The Sum and Difference of Two Cubes
1.4 - Factorizations
What is the Rule?8
8
y
y
4
4
6
6
y
y
k
k
9
9
5
5
x
x
8 8y 0y 1 4 46 06 1
y yk 0k 1
9 95x
05x 1
1.5 – Quotients
Zero Exponent0 1, as long as 0.a a
If a is a real number other than 0 and n is an integer, then
Problem:
1nn
aa
3
5
x x x x
x x x x x x
x x
x x x
x x x
2
1 1
x x x
3
5
x
x 3 5x 2x
2x2
1
x
1.5 – Quotients
Examples:
35 3
1
58x 8
1
x
47k 4
7
k 4
3
4
1
3
1
81
1 15 3 1 1
5 3
3 1 5 1
3 5 5 3
3 5
15 15
8
15
1.5 – Quotients
If a is a real number other than 0 and n is an integer, then
Examples:
1 1n nn n
a and aa a
4
1
x 0
4
x
x 0 4x 4x
6
x
x 1 6x 7x
1.5 – Quotients
Examples:5
6
y
y
5 6y 11y
9
2
r
z
2
9
z
r
26
7
2
2
6
7
2
2
7
6
49
36
1 1n nn n
a and aa a
1.5 – Quotients
¿1
𝑦11
Practice Problems
35
4
x x
x
10
45
y
y
239x
y
15
4
x x
x
16
4
x
x 16 4x 12x
10
45
y
y
10
20
y
y
10
20
y
y
10 20y 30y
410 5
1
y y
10 20
1
y y
30
1
y
2 6
2
9 x
y
2
2 69
y
x
2
681
y
x
1.5 – Quotients
¿1
𝑦30
Practice Problems
54 7a b
3 6
5 2
32
8
x y
x y
20 35a b 20
35
a
b
3 5 6 24x y 8 44x y 8
4
4x
y
3 6
5 2
32
8
x y
x y
3 5 2
6
4x x y
y
8
4
4x
y
4 5 7 5a b
1.5 – Quotients
