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CHAPTER6.1T
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à 6.1 Greatest common factor & factor by grouping
äïPlease express the following numbers in prime factored
êêform.
âS
#êêêè 48 = 2∙2∙2∙2∙3 = 2Å∙3
#êêêè 54 = 2∙3∙3∙3è= 2∙3Ä
éS
In order toïexpress the number, 48, inïprime factored form, you should
first factor it completely into a product of prime numbers.
êêêê48 = 2∙2∙2∙2∙3
êè Then express repeating factors in exponential form.
#êêêê2∙2∙2∙2∙3 = 2Å∙3
êêê This is prime factored form.
1
êêè Express 130 in prime factored form.
A)ï2∙5∙13ê B)ï4∙6∙5êïC)ï10∙13ê D) å of ç
ü
êêêë130 = 2∙65 = 2∙5∙13
Ç A
2
êêè Express 280 in prime factored form.
# A)ï13∙17êïB)ï2∙3∙3∙5∙5ë C)ï2Ä∙5∙7êD) å of ç
ü
#êêêè280 = 2∙2∙2∙5∙7 = 2Ä∙5∙7
Ç C
äïPlease find the greatest common factor.
âS
#èGiven the terms 12x and 32êè Given the terms 3xìy, 15xyì and
èthe greatest common factorêè 27xyz, the greatest common
êë is 4.êêê factor is 3xy.
éS
To find the greatest common factor of the terms 12x and 32, first express
the two terms in prime factored form.
#êêêè 12x = 2∙2∙3∙x = 2ì∙3∙x
#êêêë32 = 2∙2∙2∙2∙2 = 2É
The greatest common factor to both terms can then be identified.ïIn this
#example it is 2ì or 4.
#To find the greatest common factor of the terms 3xìy, 15xyì, and 27xyz,
first express the terms in prime factored form.
#êêêê3xìy = 3∙x∙x∙y
#êêêê15xyì = 3∙5∙x∙y∙y
êêêê27xyz = 3∙3∙3∙x∙y∙z
The greatest common factor can then be determined.ïIn 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
#êë Find the greatest common factor of 10xì and 25xy.
ëA)ï10xyê B)ï5xêïC)ï2xê D)ïå of ç
ü
#êêêê 10xì = 2∙5∙x∙x
êêêê 25xy = 5∙5∙x∙y
êêèThe greatest common factor for both is 5x.
Ç B
4
#êè Find the greatest common factor of 22aì, 14ab, and 36abì
#ëA)ï7bìêïB)ï4abê C)ï2aê D)ïå of ç
ü
#êêêë 22aìï=ï2∙11∙a∙a
êêêë 14abï=ï2∙7∙a∙b
#êêêë 36abì =ï2∙2∙3∙3∙a∙b∙b
êêë The greatest common factor is 2a.
Ç C
5
#ê Find the greatest common factor of 60pÄqìrÉ, 85pÅqÉrì
#ëA)ï5pÄqìrìëB)ï10pÉqÉëC) 2pÅqærÆè D)ïå of ç
ü
#êêêè 60pÄqìrÉï=ï2∙2∙3∙5∙pÄqìrÉ
#êêêè 85pÅqÉrìï=ï5∙17∙pÅqÉrì
#êêThe greatest common factor for both is 5pÄqìrì
Ç A
äïPlease factor the following expressions by taking out the
êêgreatest common factor.
âS
êêêë6x + 14ï=ï2(3x + 7)
#êêè 7xÄ + 14xì - 35xï=ï7x(xì + 2x - 5)
éS
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.
êêêêï6∙xï=ï2∙3∙x
êêêêè14ï=ï2∙7
The greatest common factor is 2.ïThis is written times an open set of
parençs.
êêêë 6x + 14ï=ï2∙(è )
The greatest common factor, 2, is then divided into each term and the
result placed inside the parençs.
êêêë 6x + 14 = 2(3x + 7)
#To factor the greatest common factor out of 7xÄ + 14xì - 35x, it is
#necessary to identify the greatest common factor of the terms 7xÄ, 14xì
and -35x.ïThis is seen to be "7x".
#êêë7xÄ + 14xì + (-35x)ï=ï7x(ê)
The greatest common factor is divided into each term and placed inside
the parençs.
#êêè7xÄ + 14xì + (-35x)ï=ï7x(xì + 2x - 5)
6
êëFactor out the greatest common factor forï16a - 4.
êêêêêêêè 1êè å
# A)ï2(8a - 2)êB)ï4(4a - 1)ê C)ï8(2a - ─)êD)ïof
êêêêêêêè 2êè ç
ü
êêêêè16a - 4
êêêê = 16a + (-4)
êêêê = 4(4a - 1)
Ç B
7
#êè Factor out the greatest common factor forï24mì - 15m.
êêêêêêêêêïå
A)ï8m(3m - 7)ë B)ï3m(8m - 5)êC)ï12m(2m - 3)ëD)ïof
êêêêêêêêêïç
ü
#êêêêï24mì - 15m
#êêêê= 24mì + (-15m)
êêêê= 3m(8m + (-5))
êêêê= 3m(8m - 5)
Ç B
8
#ê Factor out the greatest common factor forï3yÅ - 6yÄ + 12yì
#ê A)ï3yì(yì - 2y + 4)êê C)ï3y(yÄ - 2yì + 4y)
ê B)ï9yêêêê D)ïå of ç
ü
#êêêêï3yÅ - 6yÄ + 12yì
#êêêê= 3yÅ + (-6yÄ) + 12yì
#êêêê= 3yì(yì + (-2y) + 4)
#êêêê= 3yì(yì - 2y + 4)
Ç A
9
#êèFactor out the greatest common factor forï3xì + 8y.
#êïA)ï3x(x + 5y)êêêC)ï3(xì + 8y)
êïB)ïNo common factorêê D)ïå of ç
ü
#êêï3xì + 8yïhas no common factor other than 1.
Ç B
10êêFactor out the greatest common
#êêêfactor forï25aÅbì - 60aÄbÄ + 15aìbÅ
#ë A)ï5aìbì(5aì - 12ab + 3bì)ê C)ï5aÅbÅ(5bì - 12ab + 3aì)
#ë B)ï5ab(5aÄb - 12aìbì)êë D)ïå of ç
ü
#êêêè 25aÅbì - 60aÄbÄ + 15aìbÅ
#êêêï= 25aÅbì + (-60aÄbÄ) + 15aìbÅ
#êêêï= 5aìbì(5aì + (-12ab) + 3bì)
#êêêï= 5aìbì(5aì - 12ab + 3bì)
Ç A
äïPlease factor the following expressions by grouping.
â
#êêêë2xì + 8x + 3x + 12
#êêêè(2xì + 8x) + (3x + 12)
êêêè 2x(x + 4) + 3(x + 4)
êêêë (2x + 3)(x + 4)
éS
#In order to factor the expression, 2xì + 8x + 3x + 12, by grouping,
first insert parençs around the first two terms and the last two
terms.
#êê 2xì + 8x + 3x + 12ï=ï(2xì + 8x) + (3x + 12)
Next factor the greatest common factor out of each set of parençs.
#êë (2xì + 8x) + (3x + 12)ï=ï2x(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).
êêï2x(x + 4) + 3(x + 4)ï=ï(2x + 3)(x + 4)
This example illustrates factoring by grouping.ïIt is also possible to
factor this expression by a different grouping.
#êêêë 2xì + 8x + 3x + 12
#êêêè= (2xì + 3x) + (8x + 12)
êêêè= x(2x + 3) + 4(2x + 3)
êêêè= (x + 4)(2x + 3)
This is an equivalent answer.ïIn general an expression with four terms
can be grouped in pairs three different ways.ïIf factoring is possible,
two of ç pairings will lead to the factorization, but the third
pairing will lead to a dead end.ïIt is also important to look for a
common factor before factoring by grouping.
11
#êêèFactorï3xì + 6x + 4x + 8ïby grouping.
ê A)ï(3x + 2)(x + 4)êêïC)ï(4x + 3)(x + 2)
ê B)ï(3x + 4)(x + 2)êêïD)ïå of ç
ü
#êêêè 3xì + 6x + 4x + 8
#êêêï= (3xì + 6x) + (4x + 8)
êêêï= 3x(x + 2) + 4(x + 2)
êêêï= (3x + 4)(x + 2)
Ç B
12
#êêïFactorï2aì + 6ab + 5ab + 15bìïby grouping.
ê A)ï(5a + 2b)(b + 3)êê C)ï(2a + 5b)(a + 3b)
ê B)ï(3a + 2b)(5a + b)êêD)ïå of ç
ü
#êêêè 2aì + 6ab + 5ab + 15bì
#êêêï= (2aì + 6ab) + (5ab + 15bì)
êêêï= 2a(a + 3b) + 5b(a + 3b)
êêêï= (2a + 5b)(a + 3b)
Ç C
13
#êêèFactorï2yì + 8y - 5y - 20 by grouping.
ê A)ï(2y - 5)(y + 4)êêïC)ï(4y - 5)(y + 2)
ê B)ï(y + 5)(2y - 4)êêïD)ïå of ç
ü
#êêêè 2yì + 8y - 5y - 20
#êêêï= 2yì + 8y + (-5y) + (-20)
#êêêï= (2yì + 8y) + ((-5y) + (-20))
êêêï= 2y(y + 4) + (-5)(y + 4)
êêêï= (2y + (-5))(y + 4)
êêêï= (2y - 5)(y + 4)
Ç A
14
#êêèFactorï15rì - 12r - 10r + 8ïby grouping.
ê A)ï(4r - 3)(2r - 5)êê C)ï(3r - 2)(5r - 4)
ê B)ï(5r - 3)(2r + 4)êê D)ïå of ç
ü
#êêêè 15rì - 12r - 10r + 8
#êêêï= 15rì + (-12r) + (-10r) + 8
#êêêï= (15rì + (-12r)) + ((-10r) + 8)
êêêï= 3r(5r + (-4)) + (-2)(5r + (-4))
êêêï= (3r + (-2))(5r + (-4))
êêêï= (3r - 2)(5r - 4)
Ç C
15
#êêèFactorï3xì - 15x + x - 5èby grouping.
ê A)ï(3x + 5)(x - 1)êêïC)ï(3x + 1)(x - 5)
ê B)ï(x + 5)(3x - 1)êêïD)ïå of ç
ü
#êêêè 3xì - 15x + x - 5
#êêêï= 3xì + (-15x) + x + (-5)
#êêêï= (3xì + (-15x)) + (x + (-5))
êêêï= 3x(x + (-5)) + (x + (-5))
êêêï= (3x + 1)(x + (-5))
êêêï= (3x + 1)(x - 5)
Ç C