Section 3.1: Factors & Multiples of Whole Numbers · 1. List the multiples of each number and pick the least common between the set 2. Use prime Factorization take the greatest of

chapter 3 notes(Nancy) .notebook

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Chapter 3:

Section 3.1: Factors & Multiples of Whole Numbers

Prime Factor: a prime number that is a factor of a number.

The first 15 prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Ex: 5 is a prime factor of 50.


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Prime Factorization: the number written as a product of its prime factors.Ex: The prime factorization of 50 is

We can represent prime factorizations two ways: factor tree repeated division by prime factors

In both cases, the result is: 2 x 2 x 3 x 5 = 22 x 3 x 5. Use the method you prefer!


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Divisibility RULES:A number is divisible by

2 if it is an even number3 if the sum of the digits are divisible by 34 if the last two digits are divisible by 45 if it ends in 0 or 56 if both even and the sum of digits are divisible by 38 if the last 3 numbers are divisible by 89 if the sum of the digits are divisible by 9

Write the prime factorization of following:

a) 45 b) 36 c) 110 d) 85


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CYU pg. 135

Write the prime factorization of 2646.

Page 140 #'s 4 - 6

Greatest Common Factor (GCF): the greatest number that divides into each number in a set of numbers.Ex: 5 is the GCF of 5, 10, and 15.

You can use 2 methods to determine the GCF:1. Rainbow method2. Prime Factorization take the common factors between the set and multiply them together


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Example 2 pg. 136

Determine the greatest common factor of 138 and 198.

CYU pg. 136

Determine the greatest common factor of 126 and 144.

Page 140 #'s 8a,c,e, 9a,b


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Least Common Multiple (LCM): the least common multiple for a set of numbers.There are 2 ways to do this as well: 1. List the multiples of each number and pick the least common between the set 2. Use prime Factorization take the greatest of each number's power and multiply them all together

Example 3 pg. 137

Determine the least common multiple of 18, 20, and 30.


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CYU pg. 137

Determine the least common multiple of 28, 42, and 63.

Page 140 #'s 10a,c,e, 11a,b

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Section 3.2: Perfect Squares, Perfect Cubes, and Their Roots.

Any whole number that can be represented as the area of a square with a whole number is a perfect square.

The side length of the square is the square root of the area of the square.

We write:

25 is a perfect square and 5 is its square root.


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We can use prime factorization to determine if a number is a perfect square. If prime factors can be grouped into 2 equal groups, the number is a perfect square. Otherwise, the number is not a perfect square.

Ex: Are the following numbers perfect squares?

a) 6724 b) 1944

Example 1 pg. 144

Determine the square root of 1296.


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CYU pg. 144

Determine the square root of 1764.

Page 146 # 4

Any whole number that can be represented as the volume of a cube with a whole number edge length is a perfect cube.

The edge length of the cube is the cube root of the volume of the cube.

We write:

216 is a perfect cube and 6 is its cube root.


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We can use prime factorization to determine if a number is a perfect

cube. If prime factors can be grouped into 3 equal groups, the number is a perfect cube. Otherwise, the number is not a perfect cube.

Ex: Are the following numbers perfect cubes?

a) 13824 b) 2440


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13824 ÷ 2 = 6912

6912 ÷ 2 = 3456

3456 ÷ 2 = 1728

1728 ÷ 2 = 864

864 ÷ 2 = 432

432 ÷ 2 = 216

216 ÷ 2 = 108

108 ÷ 2 = 54

54 ÷ 2 = 27

27 ÷ 3 = 9

9 ÷ 3 = 3

(2 x 2 x 2) x (2 x 2 x 2) x (2 x 2 x 2) x (3 x 3 x 3)

Example 2 pg. 145

Determine the cube root of 1728.


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CYU pg. 145

Determine the cube root of 2744.

Page 146 #'s 5 & 6

Example 3 pg. 146

A cube has a volume of 4913 cubic inches. What is the surface area of the cube?


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CYU pg. 146

A cube has a volume 12,167 cubic feet. What is the surface area of the cube?

Page 147 #'s 7 - 10

Section 3.3: Common Factors of a Polynomial

What is the common factor in each of the following:

a) 6 and 9 b) 5 and 15 c) 12 and 16

d) 2x and 6x e) 8x and 16x f) 7x2y and 14xy4

Hint: so the GCF of the numbers but pick the lowest exponent of the common variables


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Factoring and expanding are inverse processes.After factoring, we can check by expanding.


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Example 1 pg. 152

Factor each binomial.

a) 6n + 9

b) 6c + 4c2


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CYU pg. 152

Factor each binomial

a) 3g + 6 b) 8d + 12d2

Page 155 #'s 7 & 8

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Example 2 pg. 153

Factor the trinomial 5 10z 5z2. Verify that the factors are correct (check by expanding).


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CYU pg. 153

Factor the trinomial 6 12z 18z2. Verify that the factors are correct (check by expanding).

Page 155 #'s 9 & 10

Example 3 pg. 154

Factor the trinomial 12x3y 20xy2 16x2y2. Verify that the factors are correct (check by expanding).


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CYU pg. 154

Factor the trinomial 20c4d 30c3d2 25cd. Verify that the factors are correct (check by expanding).

Page 156 #'s 15 & 16

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Activate Prior Learning:

Modelling Polynomials

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Write the polynomial represented by this set of algebra tiles.

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not a rectangle!


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State the multiplication sentence for the following


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Write a multiplication sentence with the product for each

Can you make rectangles from these polynomials? If so, what are the factors of each.

A. x2 + 2x + 1

B. y2 + 3y + 2

C. r2 + 7r + 10

D. w2 + 7w + 6

Note: These are examples with all positive terms


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Write a multiplication sentence for each. Key: Blue = +

White =

Working with negative terms.

Section 3.5: Polynomials of the Form x2 + bx + cPolynomial of degree 2.

If b, c are not 0, then there are 3 terms > trinomial

x2 + bx + c = 1x2 + bx + c

Since the leading coefficient is 1, this is a short trinomial.b is the coefficient of the second term.c is the constant term.x is the variable/unknown.


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Generally, we like polynomials to be written in descending order. That is, the term with the largest degree first and the term with the smallest degree last.

If we are given a polynomial is another order, we should rewrite it in descending order before proceeding.

The variable/unknown is not always x.

Ex: a2 + 7a 18 z2 12z + 35 4t2 16t + 128

Some variables aren't great choices: b, i, l, o, q, sWhy?

Multiplying Polynomials

When multiplying polynomials, use the distributive property.

Distributive Property: the property stating that a product can be written as a sum or difference of two products.

Ex: a(b + c) = ab + ac

Ex: (a+b)(c + d) = ac + ad + bc + bd

After expanding with the distributive property, we simplify by combining like terms.Finally, we ensure polynomial is written in descending order.


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Example 1 pg. 161

Expand and simplify.

a) (x 4)(x + 2) b) (8 k)(3 k)

CYU pg. 161

A) (c + 3)(c 7) B) ( 5 y)( 9 y)

Page 166 167 #'s 5, 9, 12, & 13


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Are the following equal?

(t 4)(t + 8) (t + 4)(t 8)

Factoring a Short Trinomial

To determine the factors of a short trinomial (x2 + bx + c), determine two integers whose product is c and whose sum is b.

These integers are the constant terms in two binomial factors, each of which has x as its first term.

Always start with the product as there are a limited number of options.

limited possibilities

infinite possibilities


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Example 2 pg. 163

Factor each trinomial

a) x2 2x 8 b) z2 12z + 35

The order in which binomial factors are written does not matter.

This is known as the commutative property.


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CYU pg. 163

A) x2 8x + 7 B) a2 + 7a 18

Page 166 167 #'s 7, 11, & 14

Hint: creating a list of factors for c helps to determine n1 and n2.

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Example 3 pg. 164

Factor

24 5d + d2


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CYU pg. 164

Page 167 #'s 15 & 17


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Sometimes, the leading coefficient is not 1. However, if it is the GCF of all 3 terms, it can be factored out.

However, it should tag along throughout the problem.

Example 4 pg. 165

Factor

4t2 16t + 128

CYU Pg. 165

Page 167 #'s 19 & 21


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Short trinomials can also be factored using algebra tiles.

https://www.youtube.com/watch?v=YAlMWv7dOqM

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Section 3.1: Factors & Multiples of Whole Numbers · 1. List the multiples of each number and pick the least common between the set 2. Use prime Factorization take the greatest of