The document discusses prime factorization and factoring monomials. It provides examples and guided practice for writing the prime factorization of numbers by decomposing them into prime factors using exponents. It also demonstrates factoring monomials by writing variables with exponents as products of the same variable.

1. Lesson 4.1, For use with pages 175-180 2. 75 ÷ 5 1. 52 ÷ 13 Please do not use a calculator to solve these.

2. Lesson 4.1, For use with pages 175-180 2. 75 ÷ 5 1. 52 ÷ 13 Please do not use a calculator to solve these. ANSWER 4 ANSWER 15

3. Divisibility Patterns and Prime Factorization Day 2 4.1

4. Vocabulary Fundamental Theorem of Arithmetic : All integers greater than one are prime or can be written as a unique product of prime numbers. Prime Factorization : the result of writing a number as a product of prime numbers. (use the factor tree method).

5. EXAMPLE 3 Writing Prime Factorization Write the prime factorization of 450 . Three factor trees are shown. Notice that each factor tree produces the same prime factorization, differing only in the order of the factors. So, 450 = 2 x 3 x 3 x 5 x 5. Using exponents, the prime factorization of 450 is 2 x 3 2 x 5 2

6. GUIDED PRACTICE for Examples 2 and 3 Tell whether the number is prime or composite . If it is composite, write its prime factorization using exponents. So, 24 = Composite; It has positive factors other then 1 and itself. 2 x 2 x 2 x 3 = 2 3 x 3 6. 24 Number : 24 Factors : 1, 2, 3, 4, 6, 8, 12, 24 24 4 6 2 2 2 3 .

7. Find the prime factorization for the following numbers. 140 27 354 34

8. Find the prime factorization for the following numbers. 140 2 2 x 5x 7 27 3 3 354 2 x 3 x 59 34 2 x 17

9. What number do these prime factorizations represent? 2 x 3 3 2 2 x 3 x 11 2 x 3 x 5 x 11

10. What number do these prime factorizations represent? 2 x 3 3 54 2 2 x 3 x 11 132 2 x 3 x 5 x 11 330

11. Vocabulary Monomial : a number, a variable, or a product of a number and one or more variables. Factor a Monomial : write it as a product of prime numbers and variables with exponents of one .

12. EXAMPLE 4 Factoring a Monomial Factor the monomial 12 x 2 y . Factor 12 . 12 x 2 y = Write x 2 as x x . = 2 2 3 x x y 2 2 3 x 2 y

13. GUIDED PRACTICE for Example 4 10. 3 mn Factor the monomial. 3 mn = 11. 18 t 2 Factor 18 . 18 t 2 = 3 m n 2 3 3 t 2 = 2 3 3 t t Write t 2 as t t .

14. GUIDED PRACTICE for Example 4 13. 54 w 3 z 4 Factor the monomial. 54 w 3 z 4 = Factor 54 . 2 3 3 3 w 3 z 4 = 2 3 3 3 w w w z z z z Write z 4 as z z z z Write w 3 as w w w = 2 3 3 3 w w w z 4

15. Homework Page #179 22-35, 52-53

16. EXAMPLE 1 Writing Factors Members of the art club are learning to do calligraphy. Their first project is to make posters to display their new lettering style. A poster will display 36 characters in order: the 26 uppercase letters of the alphabet and the digits 0 through 9 . The art club members want the characters arranged in a rectangular display with the same number of characters in each row. You can use the factors of 36 to determine how many arrangements are possible. Lettering

17. EXAMPLE 1 Writing Factors In the situation above, how many ways can the art club arrange the 36 characters in a rectangular display with rows of equal length? List the factors of 36 by writing 36 as a product of two numbers in all possible ways. 1 36 2 18 3 12 6 4 9 The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.