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

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.

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 1 12  2 6 3 4 1 4 3 2 2 3

The circled factorization is the prime factorization because all the factors are prime numbers. Factorizations of 12 1 12  2 6 3 4 1 4 3 2 2 3 A prime number has exactly two factors, itself and 1. The number 1 is not prime because it only has one factor. Remember!

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

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On Your Own! Example 1 Write the prime factorization of each number. a. 40 b. 33

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.

Example 2A: Finding the GCF of Numbers Find the GCF of each pair of numbers. 100 and 60 Method 1 List the factors. factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100 List all the factors. 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.

Example 2B: Finding the GCF of Numbers Find the GCF of each pair of numbers. 26 and 52 Method 2 Prime factorization. Write the prime factorization of each number. 26 = 2  13 52 = 2  2  13 Align the common factors. 2  13 = 26 The GCF of 26 and 52 is 26.

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On Your Own! Example 2 Find the GCF of each pair of numbers. 12 and 16 Method 1 List the factors.

On Your Own! Example 2 Find the GCF of each pair of numbers. 15 and 25 Method 2 Prime factorization.

Example 3A: Finding the GCF of Monomials Find the GCF of each pair of monomials. 15x3 and 9x2 Write the prime factorization of each coefficient and write powers as products. 15x3 = 3  5  x  x  x 9x2 = 3  3  x  x Align the common factors. 3  x  x = 3x2 Find the product of the common factors. The GCF of 3x3 and 6x2 is 3x2.

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Example 3B: Finding the GCF of Monomials Find the GCF of each pair of monomials. 8x2 and 7y3 Write the prime factorization of each coefficient and write powers as products. 8x2 = 2  2  2  x  x 7y3 = 7  y  y  y Align the common factors. The GCF 8x2 and 7y is 1. There are no common factors other than 1.

On Your Own! Example 3 Find the GCF of each pair of monomials. 18g2 and 27g3

On Your Own! Example 3b Find the GCF of each pair of monomials. 16a6 and 9b

Example 4: Factoring by Using the GCF Factor each polynomial. Check your answer. 2x2 – 4 2x2 = 2  x  x Find the GCF. 4 = 2  2 2 The GCF of 2x2 and 4 is 2. Write terms as products using the GCF as a factor. 2x2 – (2  2) 2(x2 – 2) Use the Distributive Property to factor out the GCF. Multiply to check your answer. Check 2(x2 – 2) The product is the original polynomial.  2x2 – 4

 Factor each polynomial. Check your answer. 8x3 – 4x2 – 16x 8x3 = 2  2  2  x  x  x Find the GCF. 4x2 = 2  2  x  x 16x = 2  2  2  2  x The GCF of 8x3, 4x2, and 16x is 4x. 2  2  x = 4x Write terms as products using the GCF as a factor. 2x2(4x) – x(4x) – 4(4x) Use the Distributive Property to factor out the GCF. 4x(2x2 – x – 4) Check 4x(2x2 – x – 4) Multiply to check your answer. The product is the original polynomials. 8x3 – 4x2 – 16x 

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Factor each polynomial. Check your answer. –14x – 12x2 – 1(14x + 12x2) Both coefficients are negative. Factor out –1. 14x = 2  7  x 12x2 = 2  2  3  x  x Find the GCF. The GCF of 14x and 12x2 is 2x. 2  x = 2x –1[7(2x) + 6x(2x)] Write each term as a product using the GCF. –1[2x(7 + 6x)] Use the Distributive Property to factor out the GCF. –2x(7 + 6x)

Factor each polynomial. Check your answer. 3x3 + 2x2 – 10 3x3 = 3  x  x  x Find the GCF. 2x2 = 2  x  x 10 = 2  5 There are no common factors other than 1. 3x3 + 2x2 – 10 The polynomial cannot be factored further.

On Your Own! Example 4 Factor each polynomial. Check your answer. 5b + 9b3

On Your Own! Example 4 Factor each polynomial. Check your answer. 9d2 – 82

On Your Own! Example 4 Factor each polynomial. Check your answer. –18y3 – 7y2

On Your Own! Example 4 Factor each polynomial. Check your answer. 8x4 + 4x3 – 2x2

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

Simplify each expression. 5x – 25 x2 5xy 5x = 5 x 25x2 = 5 5 x x Find the GCF. 5xy = 5 x y 5 x The GCF is 5x. 5x (1 – 5x) 5x(y) Use the Distributive Property to factor out the GCF. 5x (1 – 5x) 5x(y) Reduce and simplify. 1 - 5x y

On Your Own Example 5 Simplify each expression. a. 4x + 8 4 6a – 36a2 6ab