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2.5 Special Products
In the previous sections, the distributive property was used in multiplying
polynomials. For example, the product of two binomials bax+ and dcx+ usingthe distributive property is as follows:
( )( ) ( ) ( )( )( ) ( )( ) ( )( ) ( )( )dbcxbdaxcxax
dcxbdcxaxdcxbax
+++=+++=++
Take note that a shortcut for finding the product of two binomials is to add
the products of four pairs of terms of the binomials namely: the product of the
first terms, the productof the outer terms, the product of the inner terms, and the
productof the last terms. Thisspecial orderormethodof multiplying binomials is
known as the FOIL method. The word FOIL is formed from the first lettersof the
productsof the terms of the two binomials to be added.
The following table illustrates the FOIL method of finding the product oftwo binomials, 2x + 1 and 3x 5.
FOIL Method
F stands for the
product of the First terms
(2x+ 1) (3x 5)
( 2x) (3x) = 6x2 F
O stands for the
product of the Outer terms
(2x + 1) (3x 5 )
( 2x) (5 ) = 10x O
I stands for the
product of the Inner terms
(2x+ 1) (3x 5 )
(1) (3x) = 3x I
L stands for the
product of the Last terms
(2x + 1) (3x 5 )(1) (5) =5 L
F O I L
The product of ( ) ( ) 5310653122 +=+ xxxxx
5762 = xx Combine similar
terms
When using the FOIL method in finding certain types of products, a specificpattern is observed. Suchpatterns lead to thespecial product formulas and can be used
to find the products of binomials which are called special products. The following are the
different types of special products.
Types of Special Products
B. Product of the Sum and Difference of Two Numbers
The product of the sum and difference of two numbers is the difference of twosquares. The difference of two squares is the square of the 1st number minus the
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square of the 2nd number. Thus,
( ) ( ) 22 bababa =+
Example. Find the product of the following binomials.
( )( ) ( ) ( )
22222222
333)1 yxyxyx =+ 449 yx =
( )( ) ( ) ( ) 22232323 252525)2 srsrsr =+
=46 425 sr
( )( ) ( ) ( ) 22222222222 747474)3 cbacbacba =+
=
444
4916 cba ( )( ) ( ) ( ) 23223232 656565)4 nnnnnn yxyxyx =
=nn yx 64 3625
( ) ( ) ( )[ ] ( )
( )
( ) 162251625
454545)5
22
2
2222
++=
+=
+=+++
yxyx
yx
yxyxyx
1625502522 ++= yxyx
C. Square of a Binomial
The square of a binomial is a perfect trinomial square. A perfect trinomial
square is the square of the first number plus or minus twice the product of the two
numbers plus the square of the second number.
( ) ( ) ( )( ) ( )
( ) ( ) ( )( ) ( )222
222
2 bbaaba
bba2aba
+=
++=+
Example. Find the product of the following binomial.
( ) ( ) ( ) ( ) ( ) 222 7762676)1 yyxxyx ++=+ 22
498436 yxyx +=
( ) ( ) ( )( ) ( )222323223 4432343)2 yyxxyx ++=+ 4236 16249 yyxx +=
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( )[ ] ( )[ ] ( ) ( ) ( ) ( ) ( )
( ) ( ) 2222
15228
53121085432)5
zyxzyxyx
zzyxzyxzyxzyxzyx
++=
+=+
222
15228168 zyzxzyxyx ++=
E. Cube of a Binomial
The cube of a binomial is a quadrinomial with the following terms:1st term is the cube of the 1st number
2nd term is plus or minus thrice the square of the first number times the
second number
3rd term is plus thrice the first number times the square of the secondnumber
4th term is plus or minus the cube of the second number
( )
( ) 32233
32233
33
33
babbaaba
babbaaba
+=
+++=+
Example. Find the product of each of the following binomials.
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) 64169912274433433343)1
23
32233
+++=
+++=+
ttt
tttt
6414410827 23 +++= ttt
( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) 643629
3222322333223
8415256125
2253253525)2
yyxxyx
yyxyxxyx
+++=
+++=+
643269 860150125 yyxyxx +++=
( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) 642244266
32222222223223222
6416214912343
4473473747)3
ccbabacba
ccbacbabacba
+=
+=
642224466
64336588343 ccbacbaba +=
( )[ ] ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 25632 5441 5223235523523252)4
223223
32233
+++++++=
++++=+
bmbm bmbbmbmm
bmbmbmbm
125150756060158126223223
+++++= bmbbmmbmbbmm
F. Product of Binomialand Trinomial
The product of a binomialand a trinomialin the form )((22
babab)a + is thesum or difference of two cubes.
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3322
3322))((
ba)babb)(a(a
bababa
=++
+=++ ba
Example. Find the product of each of the following.
1)363322242 64125)4()5()162025)(45( bababbaaba +=+=++
2222 )4()4)(5()5( bbaa +
2) 27343)3()7()92149)(37(9233423623 ==++ xyxyyxxyx
222323 )3()3)(7()7( yyxx +
( )( ) ( ) ( )
( ) ( )( ) ( )22
333322
3322
2783296432)3
bbaa
bababbaaba
yyxx
yxyxyyxxyx
+
+=+=++
( )( ) ( ) ( )
( ) ( )( ) ( )2222
363322242
4433
647294343443343)4
yyxx
yxyxyxyyxxyx
++
===++
G. Product of Trinomials and Quadrinomials
The special product formulas can be used to find the products of trinomials
and quadrinomials by grouping the terms into binomials.
Example 1. Find the product of the following trinomials.
a) ( ) ( )dcadca 323323 ++
( ) ( )[ ] ( ) ( )[ ]dcadca 323323 += After grouping, the 1st term is 3a and the2ndsecond terms are (2c 3d) for both binomials. Use the formula for the
product of the sum and difference of
two numbers
( ) ( )22
323 dca = Expand ( )2
3d2c . Use the formula for
( ) ( )( ) ( )222 33262269 dda += the square of a binomial.
( )222
91249 dcdca ++= Simplify222 91249 dcdca +=
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Another solution ofExamplea:
a) ( ) ( )dcadca 323323 ++
( ) ( )[ ] ( ) ( )[ ]dcadca 323323 ++= After grouping, the 1st two
terms ( )ca 23 and( )ca 23 + are the 1st terms ofthe binomials. The 3rd terms,
3d are the 2nd terms of binomials. Use the FOIL
method
( ) ( ) ( ) ( ) ( ) ( ) 232332332332323 dcadcadcadcaca ++++= Multiply the 1st term)23)(23( ca + ca .
( ) ( ) 222 9696923 dcdadcdadca +++=
Use the formula for theproduct of the sum
and difference of twonumbers.
222 9696949 dcdadcdadca +++= Combine similar terms
22291249 dcdca +=
Note: In Example a, both the formula for the product of the sum and
difference of two numbers and the FOIL method can be used.
b) ( ) ( )pnmpnm 325525 + ( ) ( )[ ] ( ) ( )[ ]pnmpnm 325525 += After grouping, (5m + 2n)
and (5m 2n) are the 1st terms while
5p and 3p are
the 2nd terms of thebinomials. Use the
FOIL method.
( ) ( ) ( ) ( ) ( ) ( )ppnmpnmpnmnm 352532552525 +++= Multiply the 1st term
(5m+2n)(5m-2n).
( ) ( )222
15615102525 ppnpmpnpmnm ++= Use the formula for theproduct of
the sum and difference of
two numbers.
222
156151025425 ppnpmpnpmnm ++= Combine similar terms
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222 15440425 ppnpmnm ++=
Note: Unlike Example a, Example b can be expressed only as the product of
binomials after grouping. Thus, the FOIL method was used.
c) ( )2
546 zyx
( ) ( )[ ]2
546 zyx += After grouping, the 1st term is 6xand
( ) ( ) ( ) ( ) 22 5454626 zyzyxx +++= the 2nd term is (4y + 5z). Use theformula for the
square of a binomial ( square of the
difference of two numbers ).
( ) ( ) 22 54541236 zyzyxx +++= Expand (4y+5z)2. Use the formula forthesquare of
the sum of two numbers.
( ) ( ) ( ) ( ) ( ) 222 55424541236 zzyyzyxx ++++= Simplify
222 254016604836 zyzyxzxyx +++=
Note: The square of a trinomials can always be expressed as the square of a
binomial after grouping the terms of the given trinomial. Thus, the formula
for the square of a binomial is used.
Example 2. Find the product of the following quadrinomials.
a) ( )( )432432 ++++ zyxzyx
( ) ( )[ ] ( ) ( )[ ]432432 +++= zyxzyx After grouping, the 1st term is (2x+ 3y)and
the 2nd term is (z 4) for bothbinomials. Use
the formula for the product of the
sum and
difference of two numbers.( ) ( ) 22 432 += zyx Expand (2x + 3y)2 and (z - 4)2. Use
the formula
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2222 44233222 +++= zzyyxxfor the square of a binomial.
( ) ( )1689124 222 +++= zzyxyx Simplify
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1689124222 +++= zzyxyx
Note: If after grouping, the product of the given quadrinomials cannot be
expressed as the product of the sum and difference of two expressions , the
FOIL method is used.
b) ( )2
5234 + zya ( ) ( )[ ]25234 = zya After grouping, the 1st
term is (4a3y
and the 2nd term is (2z
5). Use theformula for thesquare
of a binomial.
( ) ( ) ( ) ( ) 22 525234234 += zzyaya Expand (4a 3y)2 and
(2z 5)2
. Use the formula for thesquare of a
binomial.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2222 552225234233424 +++= zzzyayyaa Multiply (4a 3y) and(2z 5). Use
FOIL method.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2222 552221 52 068233424 ++++= zzyay za zyyaa Simplify252043040121692416 222 +++++= zzyayzazyaya
Note: The square of a quadrinomial can always be expressed as the square of a
binomial after grouping the terms of the given quadrinomial. Thus the
formula for the square of a binomial is used.
G. Square of a Polynomial
The square of a polynomial is equal to the sum of the squares of each term of thepolynomial and twice the product of any combination of two terms. This method of
finding the square of a polynomial is useful if the polynomial contains more than
four terms.
Example: Find the product of the following polynomials.
1) ( )2
6357 ++ pnm
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( )( )( )( ) 3322
3322
babababa
babababa
=++
+=++
Product of Binomial and Trinomial
Square of a Polynomial
The square of a polynomial is equal to the sum of the squares of each term
of the polynomial and twice the product of any combination of two terms.
2.6 Factoring
Factoring is the reverse of multiplication. It is the process of expressing a
given polynomial as a product of its factors.
A polynomial with integral coefficients is said to be prime if it has no
monomial or polynomial factors with integral coefficients other than itself and one.
Thus, a polynomial with integral coefficients is said to be completely factoredwheneach of its polynomial factors isprime.
Types of Factoring
H. Removal of the Highest Common Factor (HCF)
A common factoris a factor contained in every term of a polynomial.
The highest common factor is the product of the greatest common factors
of the numerical coefficients and the literal coefficients having the leastexponents in every term of a polynomial.
Example: The HCF of a) 5x and 15x2is 5xb) 3ab2 and 6a is 3ac) 8x2y2z3, 16x3y3z2 and 24x4yz4 is 8x2yz2
If every term of a polynomial contains a common factor, then thepolynomial can be factored by removing the HCF. Thus, the factors are the HCF
and the quotient obtained by dividing the given polynomial by the HCF.
( )
( ) ( )dcbax
x
dxcxbxaxxdxcxbxax
+=
+=+
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Example. Factor the following completely:
+=+
xy
xyyxxyxyyx
8
5616)8(5616)1
7772
( ) ( )66 728 yxxy +=
( )
+=+
rqp
rqprqprqprqprqprqpryp
32
342533423234253342
5
2520155252015)2
( )( )22232 5435 prpqqrrqp +=
( ) ( ) ( ) ( )
( )( ) ( )
( )
=
=+
ha
hayhaxha
hayhaxahyhax
3
363
3636)3
( ) ( )yxha = 23
I. Difference of Two Squares
The factors of the difference of two squares are the sum and difference of
the square roots of the two squares.
( ) ( )bababa22 +=
Note: The sum of two squares (a2 + b2) is a prime polynomial, hence, it is notfactorable.
Example. Factor the following completely.
( )( ) ( ) ( )[ ]222
4242
3
99)1
yxb
yxbbybx
==
( )( )( )22 33 yxyxb +=
( ) ( ) ( ) ( )2222
252525)2 tratara = Factor out (5a 2)
( ) ( ) ( )trtra += 25
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( ) ( ) ( )
( )[ ] ( ) ( ) ( )[ ]22333
444
464)3
++=
=
bababa
baba
( ) ( )164424 22 +++= babababa
( )
( ) ( )[ ]33
3333
23
83243)4
mn
mnmn
yx
yxyx
=
=
( )( )mmnnmn yyxxyx 22 4223 ++=
( )
( ) ( )[ ]3232632842
523
1258337524)5
yxxy
yxxyxyxy
=
=
( )( )42222 25104523 yxyxyxxy ++=
( )
( ) ( )[ ]( )( )422422232
6667
4164242
6422128)6
bbaabaabaa
baaaba
++==
=
( )( )( )4224 416222 bbaababaa +++=
( ) ( )
( ) ( ) ( )[ ]
( )( )( )84482222
24442444
34341212)7
yyxxyxyx
yyxxyx
yxyx
+++=
++=
=
( )( )( )( )844822 yyxxyxyxyx ++++=
Note: Example 7 can also be factored as the difference of two squares
( ) ( )( )( )
( ) ( ) ( ) ( )( )( )( )( )422422422422
32323232
6666
26261212
yyxxyxyyxxyx
yxyx
yxyx
yxyx
++++=
+=
+=
=
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))()()()((
4224422422
yyxxyxyxyyxxyx +++++= D. Perfect Trinomial Square (PTS)
A trinomial is a perfect trinomial square if the first and last terms are
perfect squares and the middle term is plus or minus twice the product of the
square roots of the 1st and the last terms.The factors of a PTS are the square of the sum or the difference of
the square root of the 1st and the last terms of the given trinomial.
( )
( ) 222
222
bab2aba
bab2aba
=+
+=++
Example. Factor the following completely.
( ) ( ) ( ) ( )2222 1162611236)1 ++=+ xyxyxyyx
( )2
16 += xy
( ) ( )( ) ( ) 2222 5532325309)2 bbaababa ++=++
( )2
53 ba +=
( ) ( )( ) ( ) ( ) ( ) ( )[ ]22
22
2222
442882)3
++=
++=++
aab
aabbabba
( ) ( )2
22 += ab( ) ( )123363)4 223 +=+ aaaaaa
( ) ( )2
13 = aa
( ) ( ) ( ) ( ) 22222 2272742849)5 zzxyxyzxyzyx +=+
( ) 227 zxy =
( ) ( )( ) ( )
( )
( ) ( )[ ] 2
22
222224
22
4
442168)6
+=
=
+=+
aa
a
aaaa
( ) ( ) 22 22 += aa
( ) ( ) 2222 119 ++= xxxx
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( )
( ) ( )( ) ( )[ ]( )
( ) ( )[ ]
222
232
23232
362258
119
19
1129
1299189)7
++=
+=
++=
++=+
xxxx
xx
xxx
xxxxxx
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] 2222 5532325309)8 yxyxyxyxyxyxyxyx +++=+++ ( ) ( )[ ]
( )
( )
( )[ ] 2
2
2
2
42
82
5533
53
yx
yx
yxyx
yxyx
=
+=
++=
+=
( )2
44 yx =
E. General Trinomial
A general trinomial of the form22 )( bdyxybcadacx +++ can be factored
into the product of two binomials ))(( dycxbyax ++ . a, b, c, d can beobtained by using the trial and error method.
Trial: a) The first terms of both binomials are factors of the first term
of the given trinomial.
b) The second terms of both binomials are factors of the lastterm of the given trinomial.
To check if trialis correct, the sum of the products of the outer and inner
terms must be equal to the middle term of the given trinomial.
Example. Factor the following completely.
( ) ( )yxyxyxyx 25294845)122 +=
( )352412208)2 223 +=+ yyyyyy
( ) ( )1324 = yyy
( ) ( )72321132)3 2 ++=++ xxxx
( ) ( )174342521)4 22 += cdcdcddc
( )( )1232344)5 2 += nnnn xxxx
( ) ( ) ( )[ ] ( )[ ]5225325526)6 2 +++=++ babababa Simplify
( )( )524536 +++= baba
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( ) ( ) ( ) ( )[ ]( )[ ] ( )[ ]yxzyxz
yxyxzzyxyxzz
+=+=+
743
734321912)72222
( ) ( )yxzyxz ++= 7743
F. Factoring by Grouping
Factoring by grouping is used to factor polynomials consisting of
more than three terms.
In factoring by grouping, the terms of the given polynomial are grouped toform a binomial or a trinomial that are both factorable.
Example. Factor the following completely.
( ( )( ) ( )
( ) ( )53353
51535153)1
2
2
3232
+=++=
++=+
xx
xxx
xxxxxx
( ) ( )53 2 += xx
( ) ( )
( ) ( )
( )[ ] ( ) ( )[ ]dcbacba
dcba
dcdcbabacdabdcba
+=
=
++=++
2424
24
4481648416)2
22
22222222
( ) ( )532 += xx
( ) ( )
( ) ( )
( )[ ] ( )[ ]zyxzyx
zyx
zyxyxzyxyx
3232
32
944944)3
22
222222
+++=
+=
++=++
( ) ( )zyxzyx 3232 +++=
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( ) ( )
( ) ( )
( ) ( )[ ] ( ) ( )[ ]zyxzyx
zyx
zyzyxzyzyx
+=
=
+=+
55
5
225225)4
22
222222
( )( )zyxzyx ++= 55
( ) ( )
( ) ( )yxyx
yxyxyxyyxyx
22
244424)5
2
2222
+=
++=++
( ) ( )122 += yxyx
( ) ( )
( ) ( ) ( )bababa
babaabba
6526565
1210362512103625)6 2222
++=
+=+
( )( )26565 ++= baba
( ) ( )
( ) ( ) ( )( ) ( ) ( ) ( )yxxyxyxyx
xyxyxyxyx
xxyyxxxyyx
++=
+++=
+=+
23242
23242
638638)7
22
22
233233
( ) ( )xyxyxyx 324222 ++=
( ( ) (
( ) ( ) ( )( ) ( )[ ] 2
22
2222
32
3262
912644912644)8
yx
yxyx
yyxyxxyyxyxx
+=+++=
++++=+++
( )2
23 += yx
G. Factoring by Completing the Square
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Factoring by completing the square is applicable to binomials whose
terms are both perfect squares and to trinomials with at least two terms that
are perfect squares.The method consists of adding and subtracting a term that is a perfect
square that will make the given binomial or the trinomial a PTS. The
resulting expression is then factored as the difference of two squares.
Example. Factor the following completely.
22yxyx 4))(22(22 ++=+ ])2()[(4)1 222244 yxyx
2222
)2()2( xyyx +=
)22)(22(2222 xyyxxyyx +++=
)22)(22(2222 yxyxyxyx +++=
22
yy 44 +++=++ ])5(6)[(256)2 222224 yyyy
22 )2(])5(y10)[( 222 yy ++=
222 )2()5( yy +=
)25)(25(22 yyyy +++=
)52)(52(22 +++= yyyy
2222
baba 99 +++=++ ])2(15)6[(41536)3 2222222224 bbaabbaa 2222222 )3(])2(24)6[( abbbaa ++=
2222 )3()26( abba +=
)326)(326( 2222 abbaabba +++= )236)(236(
2222 babababa +++=
22
xx ++=+ ])3(7)[(97)4 222224 xxxx
22222 )(])3(6)[( xxx +=
222
)()3( xx =
)3)(3(22 xxxx +=
)3)(3(22 += xxxx
22
ss 44 ++=+ ])5(14)[(2514)5 222224 ssss
2222 )2(])5(s10)[( ss += 2
222 )2()5( ss =
)25)(25(22 ssss +=
)52)(52(22 += ssss
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)644
9x9x ++=+ ])6(21)[(3621 242448 xxxx
222424
)3(])6(12)[( xxx +=
2224
)3()6( xx =
)36)(36( 2424 xxxx += )63)(63(
2424 += xxxx
H. Factoring by Synthetic Division
Factoring by synthetic division is applicable to a polynomial in one variable
whose degree is higher than two.
Factor TheoremThefactor theorem is used to determine if a binomialx c is a factor of the
given polynomial.
Example 1. Determine if:
a) 1+x is a factor of 65223 + xxx
( ) 65223 += ccccP
( ) ( ) ( ) ( ) 6151211 23 +=P Replace c with1 6521 ++= Perform the indicated operations
77 +=
( ) 01 =P Since P(-1) = 0, then x + 1 is a factor of x3 + 2x2 5x 6.
b) 3y is a factor of 155323 + yyy
1553)(23 += ccccP
( ) ( ) ( ) ( ) 15353333 23 +=P Replacec with 3= 27 + 27 15 15 Perform the indicated operation
and simplify.
( ) 30543 =P
( ) 243 =P Since P(3) = 24, y 3 is not a factor of y3 + 3y2 5y 15.
Example 2. Factor the following using synthetic division.
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1
1) 343423 + xxx
1
034
34
0314
314
3434
++
+++
( ) ( ) ( )34xand1x,1x +34x3x4x
23 +
The factors of are
2) 304919234 + xxxx
25
3
011
22
0231
10155
0101321
303963
30491911
++
+++
++++
( ) ( ) ( ) ( )1xand2x,5x,3x +3049x19xxx
234 +
The factors of are
3) 2019223 + xxx
45
011
44
0431
20155
201921
++
+++
( ) ( ) ( )1xand4x,5x +
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2019x2xx23 +
The factors of are
4) 7236261322345 +++ xxxxx
3
322
021
420441
882
08421
241263
02420251
72606153
7236261321
+ ++
++++
++++
++++++
( ) ( ) ( ) ( )2xand2x,3x,3x 2 ++
7236x26x13x2xx 2345 +++ The factors of are
I. Factoring the Sum or Difference of Two Prime Odd Powers
The sum or difference of two prime odd powers can be performed using the
following formulas:
( )( )( )( )1n23n2n1nnn
1n23n2n1nnn
yyxyxxyxyx
yyxyxxyxyx
++++=
+++=+
Example. Factor the following completely.
( ) ( )
( )( )
( ) ( )16842222222
232)1
234
432234
555
+++=
+++=
+=+
xxxxx
xxxxx
xx
( ) ( )
( )( )65423324567777)2
babbabababaaba
baba
++++=
+=+
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( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( )( ) ( )6542332456
6524334256
7777
6432168422
2222222
2128)3
yxyyxyxyxyxxyx
yxyxyxyxyyxxyx
yxyx
++++++=
++++++=
=
GENERAL PROCEDURE FOR FACTORING1. Factorout any common factor.
2. If the polynomial is a binomial, factor it as the difference of two squares (a2 b2) or
thesum or difference of two cubes (a3 b3). Thesum of two squares (a2 + b2) isprime.
3. If the polynomial is a trinomial, factor it as aPTS (a2 2ab + b2) or by trial anderror method.
4. If the polynomial has more than three terms, tryfactoring by grouping.
5. If the polynomial is a binomial whose terms are both perfect squares or a trinomial
with at least two terms that are perfect squares but is not a PTS, use factoring by
completing the square.
6. If the polynomial is in one variable and has a degree higher than two, use synthetic
division
7. If the binomial is the sum or difference of two prime odd powers, apply the formulas
8. Check if all the factors are prime.
Summary of Special types of Factoring
Special Types of Factoring
Removal of the Highest Common Factor (HCF)
( ) ( )dcbaxdxcxbxax +=+Difference of Two Squares
( ) ( )bababa +=22
Sum and Difference of Two Cubes Squares
( )(( ) ( )2233
2233
babababa
babababa
+++=
++=+
Perfect Trinomial Square (PTS)
( )( )222
222
22
babababababa
=++=++