Factoring by Greatest Common Factor (GCF)



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Notes: Factoring by Greatest Common Factor (GCF)

What is a GCF??

( The largest number and/or variable that all factors of your expression have in common

How to find the GCF…

Example: Find the GCF of 15 and 30

- factors of 15 (1, 3, 5, 15) ( the largest number the have in common

- factors of 30 (1, 2, 3, 5, 6, 10, 15, 30) is 15 making it the GCF

Example: Find the GCF of two algebraic terms: 14j2k3 and 21j

A. First, determine the numeric coefficient of each term (14 and 21) and find the GCF: factors of 14 (1, 2, 7, 14) factors of 21 (1, 3, 7, 21) The two coefficients have a 7 in common; therefore the GCF of the coefficient from each term is 7

B. Next, find the GCF of the variables of the two terms:

14j2k3 = [pic]

21j = j

The two terms have one variable in common j; therefore the GCF of the variables from each term is j

C. Now, the GCF of the coefficients times the GCF of the variables is the GCF of the two terms.

(Answer: 14j2k3 and 21j have a GCF of 7j

More examples; Find the GCF of the terms:

1. 18 and 45 2. [pic] and [pic] 3. [pic] and [pic]

GCF = 9 GCF = [pic] GCF = [pic]

4. 60 and 75 5. [pic] and [pic] 6. [pic] and [pic] and [pic]

GCF = 15 GCF = [pic] GCF = [pic]

You try these; Find the GCF of the terms:

7. 25 and 100 8. 14 and 21 9. 48 and 72 and 16

10. [pic] and [pic] 11. [pic] and [pic] 12. [pic] and [pic] and [pic]

How to Factor out the GCF of the expression:

Example: 3x3 + 27x2 + 9x

1. Find the GCF of all of the terms in the expression.

A. Coefficients

3. (1, 3)

27. (1, 3, 9, 27) ( GCF of the coefficients is 3

9 (1, 3, 9)

B. Variables-

[pic]

[pic] ( GCF of the variables is [pic]

[pic]

C. Combine coefficients and variables ( GCF = 3x

2. Next, write the GCF on the left/ outside of a set of parentheses: 3x(               )

3. Then, divide each term from the original expression (3x3 + 27x2 + 9x ) by the GCF (3x), then write it inside the parenthesis. [pic] [pic] [pic]

(Factored Answer 3x(x2 + 9x + 3)

You can always check your answer by multiplying the GCF back: [pic]

*****When factoring if the GCF = 1 then it is said to be PRIME. Example: [pic] is prime*****

More examples; Factor out the GCF of the expression.

1. [pic] 2. [pic] 3. [pic] 4. [pic]

GCF = [pic] GCF = [pic] GCF = [pic] Prime

You try (with some hints): 36x2 - 64y4

1. Factor out GCF for both the coefficients and the variables

2. Divide each term in the equation by the GCF

3. Put the GCF on the outside of a set of parenthesis and the divided terms on the inside

______ ( ______ - _______)

Factor out the GCF:

1. x3 + x2 + x 2. 15a + 12b + 6c 3. x2y + 2y

4. [pic] 5. 15x2 + -50x + -10 6. 18k + 36k2 + 9k3

7. a3b2 + a3b4 + ab4 8. 18kxy + 4xy + 2k2xy 9. [pic]

10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic]

16. [pic] 17. [pic] 18. [pic]

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