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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 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)