factoring: use of the distributive property
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4.1 GCF and Factoring by Grouping
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Factoring: Use of the Distributive Property
Example 1: Find the GCF for 36 and 60.
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.
The GCF = 2 2 3 12
36 2 2 3 3
60 2 2 3 5
Step 1. First find the prime factorizations of each number.
Step 2. Circle the factors they have in common.
Answer: 2 7 14
Your Turn Problem #1
Find the GCF for 28 and 70.
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4.1 GCF and Factoring by Grouping
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Example 2: Find the GCF for x3 and x5.
Objective A: Finding the greatest common factor
3The GCF = x x x x
3x x x x
5x x x x x x
Step 1. First find the prime factorizations of each number.
Step 2. Circle the factors they have in common.
8Answer: x
Your Turn Problem #2
Find the GCF for x8 and x12.
So, when finding the GCF if variable terms, use the variable with the lowest exponent.
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4.1 GCF and Factoring by Grouping
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Example 3: Find the GCF for 84x7 and 120x3.
Objective A: Finding the greatest common factor
3The GCF = 12x
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.
5Answer: 14x
Your Turn Problem #3
Find the GCF for 42x5and 56x11.
84 2 2 3 7
120 2 2 2 3 5
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4.1 GCF and Factoring by Grouping
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Example 4: Find the GCF for (x + 5)(x - 3) and (x - 7)(x - 3).
Objective A: Finding the greatest common factor
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.
(x + 5)(x - 3)
(x - 7)(x - 3).
3xGCF :Answer
Your Turn Problem #4
Find the GCF for (x + a)(a - b) and (x + b)(a - b).
Answer: a - b
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4.1 GCF and Factoring by Grouping
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Example 5: Find the GCF for x2 – 5x – 36 and x2 +x –12
Objective A: Finding the greatest common factor
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: GCF x 4
Answer: x - 4
2x 5x 36 x 9 (x 4)
2x x 12 x 3 (x 4)
Your Turn Problem #5
2 2Find the GCF for 2x 5x 12 and 3x 13x 4.
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4.1 GCF and Factoring by Grouping
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Objective B: Factoring a Monomial from a Polynomial
General Statement ab + ac = a(b + c)The process of finding a common monomial factor is the Distributive Property in reverse.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.
1. Find the greatest common factor of these terms.
The greatest common factor of each term is 6.
Factor:Example 6. 18x 24
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.
18x 246 6
2. Divide each term by the GCF.
3. Write the answer: (GCF)(quotients of each term)
6(3x + 4)
Your Turn Problem #6
Factor: 24x 36
Answer: 12(2x 3)
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4.1 GCF and Factoring by Grouping
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5 4 4 3 3 2Factor: 6a b 18a bExamp 24l 7. ae b
1. Find the GCF for each term.GCF = 6a3b2
2. Divide each term by the GCF.5 4 4 3 3 2
3 2 3 2 3 2
6a b 18a b 24a b6a b 6a b 6a b
3 2 2 26a b a b 3ab 4
Your Turn Problem #74 4 3 3 4 5Factor: 15x y 40x y 25x y
3 3 2Answer: 5x y 3xy 8 5xy
3. Write the answer: (GCF)(quotients of each
term)
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4.1 GCF and Factoring by Grouping
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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)
Factor: Exam 2x(3a 2b) 5y(ple 8. 3a 2b)
GCF = (3a + 2b).
(3a 2b)(2x 5y)
2x(3a 2b) 5y(3a 2b)(3a 2b) (3a 2b)
2x – 5y
Your Turn Problem #8
Factor: a(2x 1) 2(2x 1)
Answer: (2x 1)(a 2)
1. Find the GCF for each term.
2. Divide each term by the GCF.
3. Write the answer: (GCF)(quotients of each
term)
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4.1 GCF and Factoring by Grouping
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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 + by1st, Factor the GCF from the first two terms and the last two terms.x(a+b)+ y(a+b)
2nd, Factor the common binomial from the expression.
Answer: (a+b)(x+y)
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. 2 Factor: 6x 9Exampl x 4x 9. ye 6y Factor out a 3x from the first
pair.3x(2x 3)Since the first term of the second pair is negative, factor out a –2y.
(22y x 3)
Lastly, factor out the common binomial.
(2x 3)(3x 2y)
Your Turn Problem #92Factor: x 4x xy 4y
Answer: (x 4)(x y)
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4.1 GCF and Factoring by Grouping
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Factor: 2bxExam cyple 10 cx. 2by Since the first two two terms do not have a common factor, we will need to rearrange the terms to factor by grouping.
2bx cx cy 2by
x(2b c) y(c 2b)
x(2b c) y(2b c)
Answer: (2b c)(x y)
Your Turn Problem #10
2Factor: 2a 3bc 2ab 3ac
Answer: (2a 3c)(a b)
The End.B.R.1-27-09
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