Algebra 1B Chapter 10 Section 1 Factors

Algebra Standard 11.0 – Students apply basic factoring techniques to second and simple third degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.

Prime and Composite Numbers Some numbers have 2 factors, the number itself and 1. This is called a prime number. Whole numbers that have more than two factors, such as 12, are called composite numbers.

Example of Prime and Composite Prime Numbers less than 20. Composite numbers less than 20 Neither Prime nor Composite 2,3,5,7,11,13,17,19 4,6,8,9,10,12,14,15,16,18 0,1

Example 1 Directions: Find the factors of each number. Then state each math problem as prime or composite number. 72 To find the factors of 72, list all pairs of whole numbers whose product is X 72, 2 X 36, 3 X 24, 4 X 18, 6 X 12, 8 X 9

The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Since 72 has more than two factors, it is a composite factor.

Your Turn Find the factor for 25. Then state each math problem as a prime or a composite number. 1 X 25 5 X 5 The factors of 25 are 1, 5, 25. Since there are more than 2 numbers it is a composite.

Example 2 Directions: Find the factors of each number. Then state each math problem as a prime or a composite. 37 There is only one pair of whole numbers whose product is X 37 The factors of 37 are 1 and 37. Therefore, 37 is a prime number.

Your Turn Directions: Find the factor of 23, and state as a prime or a composite. 1 X 23 Since there are only two factors, 23 is a prime number.

Directions: Find the factors of 79 and state each number as a prime or a composite X 79 Since 1 and 79 only have two factors, 79 is a prime number.

Directions: Find the factors of 51, and determine if it is a composite number or a prime number X 51 3 X 17 1, 3, 17, and 51 are factors of 51. Since there are more than two factors in the number 51 it is a composite number.

Example 3 Directions: Factor each monomial. 12a 2 b 4 · 3 · a · a · b What number still needs to be factored? 2 · 2 · 3 · a · a · b

Factor 100mn 3 100mn 3 50 · 2 · m · n · n · n 25 · 2 · 2 · m · n · n · n 5 · 5 · 2 · 2 · m · n · n · n

Factor 25x 2 25x 2 5 · 5 · x · x

Your Turn Factor each monomial. 15ab 2 3 · 5 · a · b · b

Factor 84xz 2 42 · 2 · x · z · z 21 · 2 · 2 · x · z · z 7 · 3 · 2 · 2 · x · z · z

Factor ⁻ 36b 3 6 · 6 · b · b · b 2 · 3 · 2 · 3 · b · b · b 2 · 2 · 3 · 3 · b · b · b

Two or more numbers may have some common prime factors. Example: 36 and = 2·2·3·3 Line up the common factors 42 = 2 · 3 · 7

The integers 36 and 42 have 2 and 3 as common prime factors. The product, which means multiplication, of these prime factors, 2 · 3 or 6. This is called the GREATEST COMMON FACTOR (GCF) of 36 and 42.

Definition of GCF Greatest common factor- The greatest common factor of two or more integers is the product of the prime factors common to the integers.

Your Definition What is the definition of Greatest Common Factor (GCF) in your own words?

Examples of GCF Directions: Find the GCF of each set of numbers or monomials. 24, 60, = 2·2·2·3 Find the prime factorization of each number. 60 = 2·2· 3·5 Line up as many common factors as possible. 72 = 2·2·2·3·3 Circle the common factors. The GCF of 24, 60, and 72 is 2·2·3 or 12.

Find the GCF of 15 and = 3 · 5 8= 2 · 2 · 2 There are no common prime factors. The only common factor is 1. So, the GCF of 15 and 8 is 1.

Find the GCF of 15a 2 b and 18ab. 15a 2 b = 3 · 5 · a · a · b 18ab = 2 · 3 · 3 · a · b The GCF of 15a 2 b and 18ab is 3 · a · b or 3ab.

Your Turn Find the GCF or each set of numbers or monomials. 75, 100, and = 3 · 5 · = 2 · 2 · 5 · 5 50 = 2 · 5 · 5 The GCF of 75, 100. and 150 is 5 · 5 or 25.

What is the GCF of 5a and 8b? 5a = 5 · a 8b = 8 · b What is the GCF when there are no common factors? Since there are no terms that are equal the GCF is 1.

What is the GCF of 24ab 2 c and 60a 2 bc? 24ab 2 c = 2 · 2 · 2 · 3 · a · b · b · c 60a 2 bc = 5 · 2 · 2 · 3 · a · a · b · c The GCF is 2 · 2 · 3 · a · b · c, which equals 12abc.

Algebra 1B Chapter 10 Section 1 Factors