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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

    =++=++


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