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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Example 2: Find the GCF for x3 and x5.


3The GCF = x x x x

3x x x x

5x x x x x x


Step 2. Circle the factors they have in common.

8Answer: x


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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Example 3: Find the GCF for 84x7 and 120x3.


3The GCF = 12x


Step 2. Circle the factors they have in common then take the variable with the lowest exponent.

5Answer: 14x


Find the GCF for 42x5and 56x11.

84 2 2 3 7

120 2 2 2 3 5

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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.

(x + 5)(x - 3)

(x - 7)(x - 3).

3xGCF :Answer


Find the GCF for (x + a)(a - b) and (x + b)(a - b).

Answer: a - b

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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: GCF x 4

Answer: x - 4

2x 5x 36 x 9 (x 4)

2x x 12 x 3 (x 4)


2 2Find the GCF for 2x 5x 12 and 3x 13x 4.

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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)


Factor: 24x 36

Answer: 12(2x 3)

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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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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


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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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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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)


2Factor: 2a 3bc 2ab 3ac

Answer: (2a 3c)(a b)

The End.B.R.1-27-09

factoring: use of the distributive property

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