38 Simplifying Expressions Using the GCF Warm Up

38 Simplifying Expressions Using the GCF Warm Up Tell whether the second number is a factor of the first number 1. 50, 6 no 2. 105, 7 yes 3. List the factors of 28. ± 1, ± 2, ± 4, ± 7, ± 14, ± 28 Tell whether each number is prime or composite. If the number is composite, write it as the product of two numbers. 4. 11 prime Holt Algebra 1 5. 98 composite; 49 2

38 Simplifying Expressions Using the GCF Learning Targets Write the prime factorization of numbers. Find the GCF of monomials. Factor polynomials by using the greatest common factor. Simplify algebraic expressions using factoring. Holt Algebra 1

38 Simplifying Expressions Using the GCF The whole numbers that are multiplied to find a product are called factors of that product. A number is divisible by its factors. You can use the factors of a number to write the number as a product. The number 12 can be factored several ways. Factorizations of 12 Holt Algebra 1

38 Simplifying Expressions Using the GCF The circled factorization is the prime factorization because all the factors are prime numbers. Factorizations of 12 Remember! A prime number has exactly two factors, itself and 1. The number 1 is not prime because it only has one factor. Holt Algebra 1

38 Simplifying Expressions Using the GCF Example 1: Writing Prime Factorizations Write the prime factorization of 98. Method 1 Factor tree Method 2 Ladder diagram Choose any two factors Choose a prime factor of 98 to begin. Keep finding to begin. Keep dividing by factors until each branch prime factors until the ends in a prime factor. quotient is 1. 98 2 98 7 49 2 49 7 7 1 98 = 2 7 7 The prime factorization of 98 is 2 7 7 or 2 72. Holt Algebra 1

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38 Simplifying Expressions Using the GCF On Your Own! Example 1 Write the prime factorization of each number. a. 40 Holt Algebra 1 b. 33

38 Simplifying Expressions Using the GCF Factors that are shared by two or more whole numbers are called common factors. The greatest of these common factors is called the greatest common factor, or GCF. Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 32: 1, 2, 4, 8, 16, 32 Common factors: 1, 2, 4 The greatest of the common factors is 4. Holt Algebra 1

38 Simplifying Expressions Using the GCF Example 2 A: Finding the GCF of Numbers Find the GCF of each pair of numbers. 100 and 60 Method 1 List the factors of 100: 1, 2, 4, 5, 10, 25, 50, 100 List all the factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 Circle the GCF. The GCF of 100 and 60 is 20. Holt Algebra 1

38 Simplifying Expressions Using the GCF Example 2 B: Finding the GCF of Numbers Find the GCF of each pair of numbers. 26 and 52 Method 2 Prime factorization. 26 = 2 13 52 = 2 2 13 = 26 Write the prime factorization of each number. Align the common factors. The GCF of 26 and 52 is 26. Holt Algebra 1

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38 Simplifying Expressions Using the GCF On Your Own! Example 2 Find the GCF of each pair of numbers. 12 and 16 Method 1 List the factors. Holt Algebra 1

38 Simplifying Expressions Using the GCF On Your Own! Example 2 Find the GCF of each pair of numbers. 15 and 25 Method 2 Prime factorization. Holt Algebra 1

38 Simplifying Expressions Using the GCF Example 3 A: Finding the GCF of Monomials Find the GCF of each pair of monomials. 15 x 3 and 9 x 2 15 x 3 = 3 5 x x x 9 x 2 = 3 3 x x 3 Write the prime factorization of each coefficient and write powers as products. Align the common factors. x x = 3 x 2 Find the product of the common factors. The GCF of 3 x 3 and 6 x 2 is 3 x 2. Holt Algebra 1

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38 Simplifying Expressions Using the GCF Example 3 B: Finding the GCF of Monomials Find the GCF of each pair of monomials. 8 x 2 and 7 y 3 8 x 2 = 2 2 2 7 y 3 = x x 7 y y y Write the prime factorization of each coefficient and write powers as products. Align the common factors. The GCF 8 x 2 and 7 y is 1. Holt Algebra 1 There are no common factors other than 1.

38 Simplifying Expressions Using the GCF On Your Own! Example 3 Find the GCF of each pair of monomials. 18 g 2 and 27 g 3 Holt Algebra 1

38 Simplifying Expressions Using the GCF On Your Own! Example 3 b Find the GCF of each pair of monomials. 16 a 6 and 9 b Holt Algebra 1

38 Simplifying Expressions Using the GCF Example 4: Factoring by Using the GCF Factor each polynomial. Check your answer. 2 x 2 – 4 2 x 2 = 2 x x 4=2 2 Find the GCF. 2 2 x 2 – (2 2) The GCF of 2 x 2 and 4 is 2. Write terms as products using the GCF as a factor. Use the Distributive Property to factor out the GCF. Multiply to check your answer. The product is the original polynomial. 2(x 2 – 2) Check 2(x 2 – 2) 2 x 2 – 4 Holt Algebra 1

38 Simplifying Expressions Using the GCF Factor each polynomial. Check your answer. 8 x 3 – 4 x 2 – 16 x 8 x 3 = 2 2 2 x x x Find the GCF. 4 x 2 = 2 2 x x 16 x = 2 2 x The GCF of 8 x 3, 4 x 2, and 16 x is 4 x. 2 2 x = 4 x Write terms as products using the GCF as a factor. 2 x 2(4 x) – x(4 x) – 4(4 x) Use the Distributive Property to 4 x(2 x 2 – x – 4) factor out the GCF. Check 4 x(2 x 2 – x – 4) Multiply to check your answer. The product is the original 8 x 3 – 4 x 2 – 16 x polynomials. Holt Algebra 1

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38 Simplifying Expressions Using the GCF Factor each polynomial. Check your answer. – 14 x – 12 x 2 – 1(14 x + 12 x 2) 14 x = 2 7 x 12 x 2 = 2 2 3 x x 2 – 1[7(2 x) + 6 x(2 x)] – 1[2 x(7 + 6 x)] – 2 x(7 + 6 x) Holt Algebra 1 Both coefficients are negative. Factor out – 1. Find the GCF. 2 The GCF of 14 x and 12 x x = 2 x is 2 x. Write each term as a product using the GCF. Use the Distributive Property to factor out the GCF.

38 Simplifying Expressions Using the GCF Factor each polynomial. Check your answer. 3 x 3 + 2 x 2 – 10 3 x 3 = 3 2 x 2 = 10 = x x x Find the GCF. 2 x x 2 5 3 x 3 + 2 x 2 – 10 There are no common factors other than 1. The polynomial cannot be factored further. Holt Algebra 1

38 Simplifying Expressions Using the GCF On Your Own! Example 4 Factor each polynomial. Check your answer. 5 b + 9 b 3 Holt Algebra 1

38 Simplifying Expressions Using the GCF On Your Own! Example 4 Factor each polynomial. Check your answer. 9 d 2 – 82 Holt Algebra 1

38 Simplifying Expressions Using the GCF On Your Own! Example 4 Factor each polynomial. Check your answer. – 18 y 3 – 7 y 2 Holt Algebra 1

38 Simplifying Expressions Using the GCF On Your Own! Example 4 Factor each polynomial. Check your answer. 8 x 4 + 4 x 3 – 2 x 2 Holt Algebra 1

38 Simplifying Expressions Using the GCF Example 5 – Simplifying Algebraic Fractions Simplify each expression. 3 p + 3 3 3 p = 3 p 3 =3 3 3 (p + 1) 3 P+1 Holt Algebra 1 Find the GCF. The GCF is 3. Use the Distributive Property to factor out the GCF. Reduce and simplify.

38 Simplifying Expressions Using the GCF Simplify each expression. 5 x – 25 x 2 5 xy 5 x = 5 x 25 x 2 = 5 5 x x 5 xy = 5 x y 5 x 5 x (1 – 5 x) 5 x(y) Holt Algebra 1 Find the GCF. The GCF is 5 x. Use the Distributive Property to factor out the GCF. 1 - 5 x y Reduce and simplify.

38 Simplifying Expressions Using the GCF On Your Own Example 5 Simplify each expression. a. 4 x + 8 4 Holt Algebra 1 b. 6 a – 36 a 2 6 ab