Download Factor Using the Greatest Common Factor

Chapter 10
Factor Using the
Greatest Common Factor
Factoring is like pressing the rewind button
on an algebra problem. You start out with an
expression and work backward so that you end
up with a multiplication problem. For example,
12 can be factored out to (6)(2). As you can see,
factoring is like unmultiplying.
There are a number of ways to factor, but the
quickest and most commonly used factoring
technique is the Greatest Common Factor (GCF)
method. The greatest common factor is the
largest term that divides evenly into each term
About Factoring
Tip
How can I determine what is the greatest
common factor of the numbers in an
expression?
in an expression. When you factor using the
greatest common factor method, you find and
remove the greatest common factor, placing
the rest of the expression in parentheses. For
example, 3 is the greatest common factor of the
expression 3 + 6 + 9, so the expression would be
factored out to 3(1 + 2 + 3).
To check your answer, work out the expression
using the distributive property. If you end up
with the original expression, you have correctly
factored the expression.
Begin by writing down all the numbers, called
factors, that evenly divide into each number in
the expression. The largest factor that all the
numbers in the expression have in common is
the greatest common factor. For instance, in
the expression 24x + 30y, write down all the
numbers that evenly divide into 24 and 30.
In this case, the greatest common factor is 6.
24 x + 30 y
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Factoring Expressions
Factor the following expressions
using the greatest common factor
method. You can check your
answers on page 262.
1) 2 + 4 + 6
2) 6 + 9 + 15
3) 12 + 18 + 60
4) 4 y + 2 x + 50 z
5) 10 a + 100 b + 50 c
6) 7 a + 7 b + 7
Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30
Factoring Out Numbers
6 x + 12 y + 15 z + 9
Factor the expression 7 a + 21 b + 35 c .
7( a + 3 b + 5 c )
Factor the expression 6 x + 12 y + 15 z + 9.
The greatest common factor is 3 .
6 x + 12 y + 15 z + 9
3
3
3
3
2x + 4y + 5z + 3
3(2 x + 4 y + 5 z + 3)
= 6 x + 12 y + 15 z + 9 Correct!
3(2 x + 4 y + 5 z + 3)
• Factoring reverses the
multiplication done
in an expression.
• When you factor an
expression, you break
the expression into
pieces, called factors,
that you can multiply
together to give you
the original expression.
1 To factor out numbers,
determine the largest
number that evenly
divides into each term
in the expression. This
number is called the
greatest common
factor, or GCF.
• In this example, the
greatest common
factor is 3.
2 Divide each term
in the expression
by the greatest
common factor.
3 Write the greatest
common factor followed
by the division result
you determined in
step 2, surrounded
by parentheses ( ).
• You have finished
factoring the expression.
4 To check your
answer, multiply
the number outside
the parentheses by
each number and
variable inside the
parentheses.
Note: The distributive property
allows you to remove a set of
parentheses by multiplying each
number and variable inside the
parentheses by a number directly
outside the parentheses. For more
information on the distributive
property, see page 30.
• If you end up with the
original expression, you
have correctly factored
the expression.
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CONTINUED
187
Chapter 10
Factor Using the Greatest
Common Factor continued
Finding the Greatest Common Factor (GCF) in an
expression requires that you find the largest term
that divides evenly into each term in the expression.
The greatest common factor method is useful when
factoring expressions that contain variables with
exponents.
In an expression that contains variables with
exponents, you need to look for variables that are
common to each term in the expression and then
choose the lowest exponent of those variables.
Tip
ctice
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Can I factor a negative variable out of an
expression?
Then write the rest of the factored
expression within parentheses. For example,
in the expression 4x 4 + 5x 2 + 3x 6, the greatest
common factor is x 2. The factored expression
would be written as x 2(4x 2 + 5 + 3x 4).
Yes. You can factor out a negative variable, such
as –a, the same way you would factor out a positive
variable, such as a. For example, to factor the
expression –a5 – a2, you can factor out –a2 by using
–a2 as the greatest common factor. Notice how the
sign (+ or –) in front of each term changes.
To check your answer, expand the new
factored expression. If the answer is the same
as the original non-factored expression, you
have solved correctly.
Factoring Expressions
Factor the following expressions
using the greatest common factor
method. You can check your
answers on page 262.
1) abc + acd + bce
2) x 2 + x 3 + x 10
3) x 2 y 3 z + x 4 y 2 z 2 + x 3 y 4
–a 5 – a 2
4) a + a 3 – a 5
–a5 – a2
–a2 –a2
5) abc 3 – bc + b 4
a3 + a0
a3 + 1
– a 2 ( a 3 + 1)
6) – b 2 – b 4 + b 7
Factoring Out Variables
a 5b 3 + a 4b 2 + a 2b
Factor the expression
5
3
4
2
2
a b + a b + a b.
a 5b 3 + a 4b 2 + a 2b
The greatest common factor (GCF) is
a 2b .
a 5b 3 + a 4b 2 + a 2b
a 2b
a 2b
a 2b
a 2 b ( a 3 b 2 + a 2 b + 1)
a 3b 2 + a 2b 1 + a 0b 0
= a 5 b 3 + a 4 b 2 + a 2 b Correct!
a 3b 2 + a 2b + 1
a 2 b ( a 3 b 2 + a 2 b + 1)
1 To factor out
variables, determine
which variable(s) each
term has in common.
• In this example, each
term has the a and b
variables in common.
2 For every variable
that each term has in
common, determine
the lowest exponent
of each variable in
the expression.
• In this example, the
lowest exponent of
the a and b variables
are a 2 and b .
Note: If a variable does not
show an exponent, assume
the exponent is 1 . For
example, b equals b 1 .
3 Write each variable with
its lowest exponent that
you determined in step 2
in alphabetical order.
These variables are the
greatest common factor,
or GCF, of the expression.
4 Divide each term in the
expression by the greatest
common factor.
Note: When you divide
variables with exponents that
have the same letter, you can
subtract the exponents. For
example, a 5 ÷ a 2 equals a 5-2 ,
which equals a 3 . A variable
with the 0 exponent equals 1.
For example, a 0 equals 1 .
5 Write the greatest
common factor
followed by the
division result you
determined in step 4,
surrounded by
parentheses ( ) .
• You have finished
factoring the
expression.
6 To check your answer,
multiply the number
outside the parentheses
by each variable and
number inside the
parentheses.
Note: When you multiply
variables with exponents
that have the same letter,
you can add the exponents.
For example, a 2 x a 3 equals
5
a 2+3 , which equals a .
• If you end up with the
original expression, you
have correctly factored
the expression.
CONTINUED
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189
Chapter 10
Factor Using the Greatest
Common Factor continued
Factoring is a lot like a game show where
contestants are given an answer and have to
come up with the question. In factoring, you
are given an expression and you must work
out the original expression.
The Greatest Common Factor (GCF) method can
be used to factor expressions that contain both
numbers and variables that have exponents. First,
determine the largest factor, or term that divides
evenly into all the other terms, that each of the
numbers in the expression has in common. Then
find the lowest exponent of any variable that
Tip
Is there a common mistake I should watch
out for?
appears in each term of the expression. Next, place
the number and variable with its exponent together to
obtain the greatest common factor for the expression.
For example, if you have the expression 3a 4 + 6a 2 + 9a 3,
the largest common factor of the numbers in front of
the variables, called coefficients, would be 3 and the
variable with the lowest exponent would be a 2.
Combining the two terms together, you arrive at a
greatest common factor of 3a 2.
As always, be sure to check your answer by expanding
out the factored expression.
One common mistake that people often
make when factoring is to forget that a term
divided by itself equals 1, not 0. If a term in an
expression is exactly the same as the greatest
common factor, when you divide the term by
the greatest common factor, you are left with
a value of 1. For example, in the expression
6x3 + 9x2 + 3x, the term 3x is divided by the
greatest common factor of 3x, so make sure
you place a 1 inside the parentheses, not a 0.
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Factoring Expressions
Factor the following expressions using
the greatest common factor method. You
can check your answers on page 262.
1) 2 x + 4 x 2 + 6 x 3
2) 10 xy + 25 x 2 + 35 xy 2
3) 8 z 3 – 16 z 5 + 20 z 6
4) 18 xyz 2 – 81 x 2 y 3 z
5) –4 a 2 – 10 a 5 + 6 a – 8
6) 9 z 3 + 7 x 2
6x 3 + 9x 2 + 3x
= 3 x (2 x 2 + 3 x + 1)
Factoring Out Numbers and Variables
24 x 5 y 2 + 30 x 3 y + 12 x 2
24 x 5 y 2 30 x 3 y 12 x 2
+
+
6x 2
6x 2
6x 2
Factor the expression 24 x
5
y 2 + 30 x 3 y + 12 x 2 .
24 x 5 y 2 + 30 x 3 y + 12 x 2
2
The greatest common factor (GCF) is 6 x .
4x 3y 2 + 5x 1y + 2x 0
6 x 2 (4 x 3 y 2 + 5 xy + 2)
4 x 3 y 2 + 5 xy + 2(1)
= 24 x 5 y 2 + 30 x 3 y + 12 x 2 Correct!
4 x 3 y 2 + 5 xy + 2
6 x 2 (4 x 3 y 2 + 5 xy + 2)
1 To factor out numbers and
variables, determine the
largest number that evenly
divides into each number
in the expression.
•
In this example, the largest
number that evenly divides
into each number in the
expression is 6 .
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2 Determine which
variable(s) each
term has in
common.
• In this example,
each term has
the x variable in
common.
3 For every variable
that each term has in
common, determine
the lowest exponent
of each variable in
the expression.
• In this example, the
lowest exponent of
the x variable is x 2 .
4 Write each variable with
its lowest exponent that
you determined in step 3
in alphabetical order.
5 Place the number you
determined in step 1 in
front of the variable(s) you
wrote down in step 4. The
number and variable(s) are
the greatest common
factor, or GCF, of the
expression.
6 Divide each term in the
expression by the greatest
common factor.
Note: When you divide variables
with exponents that have the same
letter, you can subtract the
exponents. For example, x 5 ÷ x 2
equals x 5-2 , which equals x 3 . A
variable with the 0 exponent equals
1 . For example, x 0 equals 1 . A
variable with the 1 exponent equals
itself. For example, x 1 equals x .
7 Write the greatest
common factor
followed by the
division result you
determined in step
6, surrounded by
parentheses ( ).
• You have finished
factoring the
greatest common
factor out of the
expression.
8 To check your answer,
multiply the number
and variable outside
the parentheses by
each number and
variable inside the
parentheses.
Note: When you multiply
variables with exponents
that have the same letter,
you can add the exponents.
For example, x 2 x x 3 equals
x2+3 , which equals x5 .
• If you end up with the
original expression, you
have correctly factored
the expression.
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