 294 
Е 6.1ЛGreatest common factor and factoring by grouping
ДЛPlease express the following numbers in prime factored
ИИform.
ГS

#ИИИК 48 = 2∙2∙2∙2∙3 = 2П∙3

#ИИИК 54 = 2∙3∙3∙3К= 2∙3О
ВS
In order toЛexpress the number, 48, inЛprime factored form, you should
first factor it completely into a product of prime numbers.

ИИИИ48 = 2∙2∙2∙2∙3

ИК Then express repeating factors in exponential form.

#ИИИИ2∙2∙2∙2∙3 = 2П∙3

ИИИ This is prime factored form.
 1
ИИК Express 130 in prime factored form.


 A)Л2∙5∙13И B)Л4∙6∙5ИЛC)Л10∙13И D) Ж of З
Б


ИИИЙ130 = 2∙65 = 2∙5∙13
А A
 2
ИИК Express 280 in prime factored form.


# A)Л13∙17ИЛB)Л2∙3∙3∙5∙5Й C)Л2О∙5∙7ИD) Ж of З

Б


#ИИИК280 = 2∙2∙2∙5∙7 = 2О∙5∙7
А C
ДЛPlease find the greatest common factor.
ГS

#КGiven the terms 12x and 32ИК Given the terms 3xНy, 15xyН and
Кthe greatest common factorИК 27xyz, the greatest common
ИЙ is 4.ИИИ factor is 3xy.
ВS
To find the greatest common factor of the terms 12x and 32, first express
the two terms in prime factored form.

#ИИИК 12x = 2∙2∙3∙x = 2Н∙3∙x
#ИИИЙ32 = 2∙2∙2∙2∙2 = 2Р

The greatest common factor to both terms can then be identified.ЛIn this
#example it is 2Н or 4.

#To find the greatest common factor of the terms 3xНy, 15xyН, and 27xyz,
first express the terms in prime factored form.

#ИИИИ3xНy = 3∙x∙x∙y
#ИИИИ15xyН = 3∙5∙x∙y∙y
ИИИИ27xyz = 3∙3∙3∙x∙y∙z

The greatest common factor can then be determined.ЛIn this case it is
3∙x∙y

The greatest common factor can be described as the largest expression
that can be divided evenly into each term.
 3
#ИЙ Find the greatest common factor of 10xН and 25xy.


ЙA)Л10xyИ B)Л5xИЛC)Л2xИ D)ЛЖ of З
Б
#ИИИИ 10xН = 2∙5∙x∙x
ИИИИ 25xy = 5∙5∙x∙y

ИИКThe greatest common factor for both is 5x.
А B
 4
#ИК Find the greatest common factor of 22aН, 14ab, and 36abН


#ЙA)Л7bНИЛB)Л4abИ C)Л2aИ D)ЛЖ of З
Б
#ИИИЙ 22aНЛ=Л2∙11∙a∙a
ИИИЙ 14abЛ=Л2∙7∙a∙b
#ИИИЙ 36abН =Л2∙2∙3∙3∙a∙b∙b

ИИЙ The greatest common factor is 2a.
А C
 5
#И Find the greatest common factor of 60pОqНrР, 85pПqРrН


#ЙA)Л5pОqНrНЙB)Л10pРqРЙC) 2pПqСrТК D)ЛЖ of З
Б
#ИИИК 60pОqНrРЛ=Л2∙2∙3∙5∙pОqНrР
#ИИИК 85pПqРrНЛ=Л5∙17∙pПqРrН

#ИИThe greatest common factor for both is 5pОqНrН
А A
ДЛPlease factor the following expressions by taking out the
ИИgreatest common factor.
ГS

ИИИЙ6x + 14Л=Л2(3x + 7)

#ИИК 7xО + 14xН - 35xЛ=Л7x(xН + 2x - 5)
ВS
In order to factor 6x + 14 by taking out the greatest common factor, it
is first necessary to identify the greatest common factor of the terms
6x and 14.
ИИИИЛ6∙xЛ=Л2∙3∙x
ИИИИК14Л=Л2∙7

The greatest common factor is 2.ЛThis is written times an open set of
parenЗs.
ИИИЙ 6x + 14Л=Л2∙(К )

The greatest common factor, 2, is then divided into each term and the
result placed inside the parenЗs.
ИИИЙ 6x + 14 = 2(3x + 7)

#To factor the greatest common factor out of 7xО + 14xН - 35x, it is
#necessary to identify the greatest common factor of the terms 7xО, 14xН
and -35x.ЛThis is seen to be '7x'.

#ИИЙ7xО + 14xН + (-35x)Л=Л7x(И)

The greatest common factor is divided into each term and placed inside
the parenЗs.
#ИИК7xО + 14xН + (-35x)Л=Л7x(xН + 2x - 5)
 6
ИЙFactor out the greatest common factor forЛ16a - 4.

ИИИИИИИК 1ИК Ж
 A)Л2(8a - 2)ИB)Л4(4a - 1)И C)Л8(2a - ─)ИD)Лof
ИИИИИИИК 2ИК З
Б

ИИИИК16a - 4
ИИИИ = 16a + (-4)
ИИИИ = 4(4a - 1)

А B
 7
#ИК Factor out the greatest common factor forЛ24mН - 15m.

ИИИИИИИИИЛЖ
 A)Л8m(3m - 7)Й B)Л3m(8m - 5)ИC)Л12m(2m - 3)ЙD)Лof
ИИИИИИИИИЛЗ
Б
#ИИИИЛ24mН - 15m
#ИИИИ= 24mН + (-15m)
ИИИИ= 3m(8m + (-5))
ИИИИ= 3m(8m - 5)
А B
 8
#И Factor out the greatest common factor forЛ3yП - 6yО + 12yН


#И A)Л3yН(yН - 2y + 4)ИИ C)Л3y(yО - 2yН + 4y)
И B)Л9yИИИИ D)ЛЖ of З
Б
#ИИИИЛ3yП - 6yО + 12yН
#ИИИИ= 3yП + -6yО) + 12yН
#ИИИИ= 3yН(yН + (-2y) + 4)
#ИИИИ= 3yН(yН - 2y + 4)
А A
 9
#ИКFactor out the greatest common factor forЛ3xН + 8y.


#ИЛA)Л3x(x + 5y)ИИИC)Л3(xН + 8y)
ИЛB)ЛNo common factorИИ D)ЛЖ of З
Б


#ИИЛ3xН + 8yЛhas no common factor other than 1.

А B
 10ИИFactor out the greatest common
#ИИИfactor forЛ25aПbН - 60aОbО + 15aНbП


#Й A)Л5aНbН(5aН - 12ab + 3bН)И C)Л5aПbП(5bН - 12ab + 3aН)
#Й B)Л5ab(5aОb - 12aНbН)ИЙ D)ЛЖ of З
Б
#ИИИК 25aПbН - 60aОbО + 15aНbП
#ИИИЛ= 25aПbН + (-60aОbО) + 15aНbП
#ИИИЛ= 5aНbН(5aН + (-12ab) + 3bН)
#ИИИЛ= 5aНbН(5aН - 12ab + 3bН)
А A
ДЛPlease factor the following expressions by grouping.
Г
#ИИИЙ2xН + 8x + 3x + 12
#ИИИК(2xН + 8x) + (3x + 12)
ИИИК 2x(x + 4) + 3(x + 4)
ИИИЙ (2x + 3)(x + 4)
ВS
#In order to factor the expression, 2xН + 8x + 3x + 12, by grouping,
first insert parenЗs around the first two terms and the last two
terms.
#ИИ 2xН + 8x + 3x + 12Л=Л(2xН + 8x) + (3x + 12)

Next factor the greatest common factor out of each set of parenЗs.
#ИЙ (2xН + 8x) + (3x + 12)Л=Л2x(x + 4) + 3(x + 4)

Finally since (x + 4) is a common factor of the terms 2x(x + 4) and
3(x + 4), it can be factored out of the expression 2x(x + 4) + 3(x + 4).

ИИЛ2x(x + 4) + 3(x + 4)Л=Л(2x + 3)(x + 4)

This example illustrates factoring by grouping.ЛIt is also possible to
factor this expression by a different grouping.

#ИИИЙ 2xН + 8x + 3x + 12
#ИИИК= (2xН + 3x) + (8x + 12)
ИИИК= x(2x + 3) + 4(2x + 3)
ИИИК= (x + 4)(2x + 3)

This is an equivalent answer.ЛIn general an expression with four terms
can be grouped in pairs three different ways.ЛIf factoring is possible
two of З pairings will lead to the factorization, but the third
pairing will lead to a dead end.ЛIt is also important to look for a
common factor before factoring by grouping.
 11
#ИИКFactorЛ3xН + 6x + 4x + 8Лby grouping.


И A)Л(3x + 2)(x + 4)ИИЛC)Л(4x + 3)(x + 2)
И B)Л(3x + 4)(x + 2)ИИЛD)ЛЖ of З
Б
#ИИИК 3xН + 6x + 4x + 8
#ИИИЛ= (3xН + 6x) + (4x + 8)
ИИИЛ= 3x(x + 2) + 4(x + 2)
ИИИЛ= (3x + 4)(x + 2)
А B
 12
#ИИЛFactorЛ2aН + 6ab + 5ab + 15bНЛby grouping.


И A)Л(5a + 2b)(b + 3)ИИ C)Л(2a + 5b)(a + 3b)
И B)Л(3a + 2b)(5a + b)ИИD)ЛЖ of З
Б
#ИИИК 2aН + 6ab + 5ab + 15bН
#ИИИЛ= (2aН + 6ab) + (5ab + 15bН)
ИИИЛ= 2a(a + 3b) + 5b(a + 3b)
ИИИЛ= (2a + 5b)(a + 3b)
А C
 13
#ИИКFactorЛ2yН + 8y - 5y - 20 by grouping.


И A)Л(2y - 5)(y + 4)ИИЛC)Л(4y - 5)(y + 2)
И B)Л(y + 5)(2y - 4)ИИЛD)ЛЖ of З
Б
#ИИИК 2yН + 8y - 5y - 20
#ИИИЛ= 2yН + 8y + (-5y) + (-20)
#ИИИЛ= (2yН + 8y) + ((-5y) + (-20))
ИИИЛ= 2y(y + 4) + (-5)(y + 4)
ИИИЛ= (2y + (-5))(y + 4)
ИИИЛ= (2y - 5)(y + 4)
А A
 14
#ИИКFactorЛ15rН - 12r - 10r + 8Лby grouping.


И A)Л(4r - 3)(2r - 5)ИИ C)Л(3r - 2)(5r - 4)
И B)Л(5r - 3)(2r + 4)ИИ D)ЛЖ of З
Б
#ИИИК 15rН - 12r - 10r + 8
#ИИИЛ= 15rН + (-12r) + (-10r) + 8
#ИИИЛ= (15rН + (-12r)) + ((-10r) + 8)
ИИИЛ= 3r(5r + (-4)) + (-2)(5r + (-4))
ИИИЛ= (3r + (-2))(5r + (-4))
ИИИЛ= (3r - 2)(5r - 4)
А C
 15
#ИИКFactorЛ3xН - 15 + x - 5Кby grouping.


И A)Л(3x + 5)(x - 1)ИИЛC)Л(3x + 1)(x - 5)
И B)Л(x + 5)(3x - 1)ИИЛD)ЛЖ of З
Б
#ИИИК 3xН - 15x + x - 5
#ИИИЛ= 3xН + (-15x) + x + (-5)
#ИИИЛ= (3xН + (-15x)) + (x + (-5))
ИИИЛ= 3x(x + (-5)) + (x + (-5))
ИИИЛ= (3x + 1)(x + (-5))
ИИИЛ= (3x + 1)(x - 5)
А C