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. 1
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 = 2 2 x 3 x 5. Use the method you prefer! 2
Divisibility RULES: A number is divisible by 2 if it is an even number 3 if the sum of the digits are divisible by 3 4 if the last two digits are divisible by 4 5 if it ends in 0 or 5 6 if both even and the sum of digits are divisible by 3 8 if the last 3 numbers are divisible by 8 9 if the sum of the digits are divisible by 9 Write the prime factorization of following: a) 45 b) 36 c) 110 d) 85 3
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 method 2. Prime Factorization take the common factors between the set and multiply them together 4
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 5
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. 6
CYU pg. 137 Determine the least common multiple of 28, 42, and 63. Page 140 #'s 10a,c,e, 11a,b?? 7
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. 8
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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. 10
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. 11
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 12
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. 13
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? 14
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 Hint: so the GCF of the numbers but pick the lowest exponent of the common variables d) 2x and 6x e) 8x and 16x f) 7x 2 y and 14xy 4 15
Factoring and expanding are inverse processes. After factoring, we can check by expanding. 16
Example 1 pg. 152 Factor each binomial. a) 6n + 9 b) 6c + 4c 2 17
CYU pg. 152 Factor each binomial a) 3g + 6 b) 8d + 12d 2 Page 155 #'s 7 & 8 18
Example 2 pg. 153 Factor the trinomial 5 10z 5z 2. Verify that the factors are correct (check by expanding). 19
CYU pg. 153 Factor the trinomial 6 12z 18z 2. Verify that the factors are correct (check by expanding). Page 155 #'s 9 & 10 Example 3 pg. 154 Factor the trinomial 12x 3 y 20xy 2 16x 2 y 2. Verify that the factors are correct (check by expanding). 20
CYU pg. 154 Factor the trinomial 20c 4 d 30c 3 d 2 25cd. Verify that the factors are correct (check by expanding). Page 156 #'s 15 & 16 21
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Activate Prior Learning: Modelling Polynomials Write the polynomial represented by this set of algebra tiles. not a rectangle! 23
State the multiplication sentence for the following 24
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. x 2 + 2x + 1 Note: These are examples with all positive terms B. y 2 + 3y + 2 C. r 2 + 7r + 10 D. w 2 + 7w + 6 25
Working with negative terms. Write a multiplication sentence for each. Key: Blue = + White = Section 3.5: Polynomials of the Form x 2 + bx + c Polynomial of degree 2. If b, c are not 0, then there are 3 terms > trinomial x 2 + bx + c = 1x 2 + 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. 26
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: a 2 + 7a 18 z 2 12z + 35 4t 2 16t + 128 Some variables aren't great choices: b, i, l, o, q, s Why? 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. 27
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 28
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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 (x 2 + bx + c), determine two integers whose product is c and whose sum is b. limited possibilities Always start with the product as there are a infinite possibilities limited number of options. These integers are the constant terms in two binomial factors, each of which has x as its first term. 30
Example 2 pg. 163 Factor each trinomial a) x 2 2x 8 b) z 2 12z + 35 The order in which binomial factors are written does not matter. This is known as the commutative property. 31
CYU pg. 163 A) x 2 8x + 7 B) a 2 + 7a 18 Hint: creating a list of factors for c helps to determine n 1 and n 2. Page 166 167 #'s 7, 11, & 14 32
Example 3 pg. 164 Factor 24 5d + d 2 33
CYU pg. 164 Page 167 #'s 15 & 17 34
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 4t 2 16t + 128 CYU Pg. 165 Page 167 #'s 19 & 21 35
Short trinomials can also be factored using algebra tiles. https://www.youtube.com/watch?v=yalmwv7doqm 36