5.1a S KM & PP 1

Factoring Polynomials:The G.C.F.

5.1a S KM & PP 2

What is a FACTOR?

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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5.1a S KM & PP 3

What are the Factors of 12?

121 12 1 x 12 = 12

122 6 2 x 6 = 12

123 4 3 x 4 = 12

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

5.1a S KM & PP 4

What are the Factors of 20?

201 20 1 x 20 = 20

202 10 2 x 10 = 20

204 5 4 x 5 = 20

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

5.1a S KM & PP 5

What is a COMMON FACTOR?

If two or more numbers or expressions have the

same factor, that factor is called a COMMON

FACTOR.

xmb ynb

5.1a S KM & PP 6

What if the only COMMON FACTOR

is 1 ?

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

1,we say the numbers or

expressions are:

5.1a S KM & PP 7

What are the COMMON FACTORS of 16 and

27?

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

Factors of 27: 1, 3, 9, 27

1

27

2

98

3

16

4

16 and 27 are RELATIVELY PRIME

5.1a S KM & PP 8

What are the COMMON FACTORS of 12 and

20?

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

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

1

20

2

104

5

612

3

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

5.1a S KM & PP 9

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

What is theGreatest Common Factor

of 12 and 20?

The Greatest Common Factor of 12 and 20 is 4.

The largest number in the list ofcommon factors is 4.

5.1a S KM & PP 10

Factors of x3: 1, x1, x2, x3

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

1x1

x5

x3

x4

x2

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

What are theCommon Factors

of x3 and x5?

5.1a S KM & PP 11

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

What is theGreatest Common Factor

of x3 and x5?

The Greatest Common Factor of x3 and x5 is x3.

The largest number (disregarding the sign) in

the list of common factors is x3.

5.1a S KM & PP 12

What is theGreatest Common Factor

of xn and xm?

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.

5.1a S KM & PP 13

What is theGreatest Common Factor

of 12x3 and 20x5?

The Greatest Common Factor of x3 and x5 is x3.

The Greatest Common Factor of 12 and 20 is 4.

The Greatest Common Factor of 12x3 and 20x5 is

4x3.

5.1a S KM & PP 14

Finding theGreatest Common Factor“The Algebraic Approach”

To Determine the GCF:

1) Factor each expression completely

2) The GCF is the product of the common bases each raised to the smallest exponent

Let’s try a few examples!

5.1a S KM & PP 15

Let’s Build theGreatest Common Factor

of 12x3 and 20x5

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

or

3x12

5x20

32 x32

52 x52

32 x2 3x4

5.1a S KM & PP 16

Let’s Build theGreatest Common Factor

of 90x2y3z and 50y4z5

The GCF of 90x2y3z and 50y4z5 is the product of the “common” bases raised to the smallest exponent.

or

zyx90 32

54zy50

zyx532 322

542 zy52

zy52 3 zy10 3

5.1a S KM & PP 17

Let’s Build theGreatest Common Factor

of 21xy5z and 30y4

The GCF of 21xy5z and 30y4 is the product of the “common” bases raised to the smallest exponent.

or

zxy21 5

4y30

zxy73 5

4y532

4y3 4y3

5.1a S KM & PP 18

Let’s Build theGreatest Common Factor

of 21x2z and 10y4

21x2z and 10y4 have no common factors!

The only factor common to both expressions is 1.

21x2z and 10y4 are RELATIVELY PRIME

because their GCF is 1.

zx21 2

4y10

zx73 2

4y52

5.1a S KM & PP 19

What does it mean to FACTOR?

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

FACTORED .

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5.1a S KM & PP 20

Factoring out the GCFis Reversing

the Distributive Property

5.1a S KM & PP 21

Factor out the GCFfrom 12x5 + 20x3

12x5 20x3

4x3 4x33x2 5

4x3 3x2 5

5.1a S KM & PP 22

Factor: 12x5 + 20x3

So, by reversing the Distributive Property,

the binomial

4x3(3x2 + 5)

can be written as the product

12x5 + 20x3

12x5 + 20x3= 4x3(3x2 + 5)

5.1a S KM & PP 23

Factor: 24x3 + 18x

= 6x(4x2 + 3)

The GCF of 24 and 18 is 6.

The GCF of x3 and x is x.

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

24x3 + 18x= 6x(4x2) + 6x(3)

5.1a S KM & PP 24

Factor: 17x4 - 8x3

= x3(17x - 8)

The GCF of 17 and 8 is 1.

The GCF of x4 and x3 is x3.

The GCF of 17x4 and x3 is 1x3 or

x3.

17x4 - 8x3 = x3(17x) – x3(8)

5.1a S KM & PP 25

Factor: 6x2y5 - 8x3y4

GCF of 6 and 8 is 2

GCF of x2y5 and x3y4 is x2y4

GCF of 6x2y5 and 8x3y4 is 2x2y4

6x2y5 - 8x3y4

= 2x2y4(3y) –2x2y4(4x)

= 2x2y4(3y - 4x)

5.1a S KM & PP 26

Factor:9x3 - 12x2 + 3x

GCF of 9, 12 and 3 is 3

GCF of x3, x2 and x is x

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

9x3 - 12x2 + 3x

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

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

5.1a S KM & PP 27

Factor:9x3 – 11y2 + 3

GCF of 9, 11 and 3 is 1

GCF of x3 and y2 is 1

GCF of 9x3, 11y2 and 3 is 1

9x3 – 11y2 + 3

=1(9x3 – 11y2 + 3)

5.1a S KM & PP 28

Factor a Tricky One!

x(x + 2) – 6(x + 2)

x(x + 2) – 6(x + 2)

= ( x + 2 )( x – 6 )

5.1a S KM & PP 29

Another Tricky One!

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

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

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

5.1a S KM & PP 30

That’s All for Now!