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Factoring: Use of the Distributive Property

Your Turn Problem #1. Find the GCF for 28 and 70. Factoring: Use of the Distributive Property. Objective A: Finding the greatest common factor.

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Factoring: Use of the Distributive Property

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  1. Your Turn Problem #1 Find the GCF for 28 and 70. Factoring: Use of the Distributive Property Objective A: Finding the greatest common factor The greatest common factor is the largest number that divides evenly into a set of numbers. For example, the GCF of 12 and 18 would be 6 because 6 is the largest number that divides evenly into both numbers. Example 1: Find the GCF for 36 and 60. Step 1. First find the prime factorizations of each number. Step 2. Circle the factors they have in common.

  2. Your Turn Problem #2 Find the GCF for x8 and x12. Objective A: Finding the greatest common factor Example 2: Find the GCF for x3 and x5. Step 1. First find the prime factorizations of each number. Step 2. Circle the factors they have in common. So, when finding the GCF if variable terms, use the variable with the lowest exponent.

  3. Your Turn Problem #3 Find the GCF for 42x5and 56x11. Objective A: Finding the greatest common factor Example 3: Find the GCF for 84x7 and 120x3. Step 1. First find the prime factorizations of each number. Step 2. Circle the factors they have in common then take the variable with the lowest exponent.

  4. (x + 5)(x - 3) (x - 7)(x - 3). Your Turn Problem #4 Find the GCF for (x + a)(a - b) and (x + b)(a - b). Objective A: Finding the greatest common factor Example 4: Find the GCF for (x + 5)(x - 3) and (x - 7)(x - 3). Step 1. Write the product of each. Step 2. Circle the factors they have in common. In this case, the common factor is a binomial. Answer: a - b

  5. Your Turn Problem #5 Objective A: Finding the greatest common factor Example 5: Find the GCF for x2 – 5x – 36 and x2 +x –12 Step 1. Write factored form of each. Step 2. Circle the factors they have in common. In this case, the common factor is a binomial. Answer: x - 4

  6. Objective B: Factoring a Monomial from a Polynomial The process of finding a common monomial factor is the Distributive Property in reverse. General Statement ab + ac = a(b + c) Procedure: To factor a monomial from a polynomial Step 1. Find the greatest common factor (GCF) of all terms of the polynomial. Step 2. Divide each term by this GCF. Step 3. Write the answer in the form: (GCF)(quotients of each term). Note: Steps 2 and 3 are the Distributive Property worked backwards. Notes: 1. The greatest common factor of two or more integers is the greatest integer that is a common factor of all the integers. 2. The greatest common factor of variable factors is the smallest exponent of each variable that is common to all. Your Turn Problem #6 1. Find the greatest common factor of these terms. The greatest common factor of each term is 6. 2. Divide each term by the GCF. 6(3x + 4) 3. Write the answer: (GCF)(quotients of each term)

  7. Your Turn Problem #7 1. Find the GCF for each term. GCF = 6a3b2 2. Divide each term by the GCF. • Write the answer: • (GCF)(quotients of each term)

  8. Objective C: Factoring a Binomial from a Polynomial (The greatest common factor is a binomial) General Statement a(x+y) +b(x+y) = (x+y)(a+b) Your Turn Problem #8 1. Find the GCF for each term. GCF = (3a + 2b). 2. Divide each term by the GCF. 2x – 5y • Write the answer: • (GCF)(quotients of each term)

  9. Objective D: Factoring by Grouping Some polynomials can be factored by grouping terms in such a way that a common binomial factor is found. Example: ax + bx + ay + by Your Turn Problem #9 1st, Factor the GCF from the first two terms and the last two terms. x(a+b)+ y(a+b) Answer:(a+b)(x+y) 2nd, Factor the common binomial from the expression. Notes: The goal is to obtain a common binomial in both terms. Sometimes the order of the polynomial may have to be rearranged to achieve the desired outcome. If the first term of the second pair is negative, factor out the negative along with the GCF. Factor out a 3x from the first pair. Since the first term of the second pair is negative, factor out a –2y. Lastly, factor out the common binomial.

  10. Answer: Your Turn Problem #10 The End. B.R. 1-27-09 Since the first two two terms do not have a common factor, we will need to rearrange the terms to factor by grouping.

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