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Factoring Polynomials: The G.C.F.

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Factoring Out The

Greatest Common Factor

C

G

F

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product

factor

factor

a and b are factors of c

A FACTOR is a number or an expression that is multiplied with at least one other number or expression to form a PRODUCT.

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1

1 x 12 = 12

12

12

2 x 6 = 12

2

6

12

3

3 x 4 = 12

4

12

Factors of 12: 1, 2, 3, 4, 6, 12

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1

1 x 20 = 20

20

20

2 x 10 = 20

2

10

20

4

4 x 5 = 20

5

20

Factors of 20: 1, 2, 4, 5, 10, 20

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If

and

then b is a "common factor" of x and y.

If two or more numbers or expressions have the same factor, that factor is called a COMMON FACTOR.

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If the only common factor of two or more numbers or expressions is the number 1,

we say the numbers or expressions are:

Relatively Prime

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Factors of 16: 1, 2, 4, 8, 16

Factors of 27: 1, 3, 9, 27

1

2

3

4

9

8

16

27

16 and 27 are RELATIVELY PRIME

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Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 20: 1, 2, 4, 5, 10, 20

1

2

5

3

10

4

6

20

12

Common Factors of 12 and 20: 1, 2, 4

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Common Factors of 12 and 20: 1, 2, 4

The largest number in the list ofcommon factors is 4.

The Greatest Common Factor of 12 and 20 is 4.

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Factors of x3: 1, x1, x2, x3

Factors of x5: 1, x1, x2, x3,x4, x5

1

x4

x1

x2

x5

x3

Common Factors of x3 and x5: 1, x1, x2, x3

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Common Factors of x3 and x5: 1, x1, x2, x3

The largest number (disregarding the sign) in the list of common factors is x3.

The Greatest Common Factor of x3 and x5 is x3.

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TheGreatest Common Factor of xn and xm is the term that has the smaller exponent.

GCF of x7 and x2 is x2.

GCF of x4 and x8 is x4.

GCF of x and x3 is x.

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The Greatest Common Factor of 12 and 20 is 4.

The Greatest Common Factor of x3 and x5 is x3.

The Greatest Common Factor of 12x3 and 20x5 is 4x3.

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To Determine the GCF:1) Factor each expression completely2) The GCF is the product of the common bases each raised to the smallest exponent

Let’s try a few examples!

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The GCF of 12x3 and 20x5 is the product of the “common” bases raised to the smallest exponent.or

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The GCF of 90x2y3zand 50y4z5 is the product of the “common” bases raised to the smallest exponent.or

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The GCF of 21xy5zand 30y4 is the product of the “common” bases raised to the smallest exponent.or

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21x2zand 10y4 have no common factors!The only factor common to both expressions is 1. 21x2zand 10y4 are RELATIVELY PRIMEbecause their GCF is 1.

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factor

factor

c has been factored into the product of a and b

When a number, expression, or polynomial is rewritten as a product of factors, we say it has been FACTORED .

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+

+

(

)

+

=

Now let's try an example!

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12x5

+

20x3

GCF

GCF

+

5

4x3

3x2

4x3

GCF

(

)

4x3

+

3x2

5

=

Reverse Distributive Property

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So, by reversing the Distributive Property, the binomial

12x5 + 20x3

can be written as the product

4x3(3x2 + 5)

12x5 + 20x3

= 4x3(3x2 + 5)

product

sum

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The GCF of 24 and 18 is 6.The GCF of x3 and x is x.The GCF of 24x3 and 18x is 6x.

sum

GCF

GCF

24x3 + 18x

= 6x(4x2) + 6x(3)

GCF

= 6x(4x2 + 3)

product

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The GCF of 17 and 8 is 1.The GCF of x4 and x3 is x3.The GCF of 17x4 and x3 is 1x3 orx3.

sum

GCF

GCF

= x3(17x) – x3(8)

17x4 - 8x3

GCF

= x3(17x - 8)

product

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GCF of 6 and 8 is 2GCF of x2y5 and x3y4 is x2y4GCF of 6x2y5 and 8x3y4 is 2x2y4

sum

6x2y5- 8x3y4

GCF

GCF

= 2x2y4(3y) –2x2y4(4x)

GCF

= 2x2y4(3y - 4x)

product

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GCF of 9, 12 and 3 is 3GCF of x3, x2 and x is xGCF of 9x3, 12x2 and 3x is 3x

sum

9x3 - 12x2 + 3x

GCF

GCF

GCF

= 3x(3x2) - 3x(4x)+ 3x(1)

GCF

= 3x(3x2 - 4x + 1)

product

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GCF of 9, 11 and 3 is 1GCF of x3 and y2 is 1GCF of 9x3, 11y2 and 3 is 1

9x3 – 11y2 + 3

Prime or Nonfactorable

so leave it as is!

=1(9x3 – 11y2 + 3)

shown with the GCF factored out

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x(x + 2) – 6(x + 2)

GCF

GCF

x(x + 2) – 6(x + 2)

GCF

= ( x + 2 )( x – 6 )

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(x-7)3x + (x-7)5

GCF

GCF

(x - 7)3x +(x - 7)5

GCF

= ( x - 7 )( 3x + 5 )

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That’s All for Now!

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