ร12-64

1 6x3 −11x2 + 6x = 1

6x3 −11x2 + 6x = 1

6x3 −11x2 + 6x −1 = 0

p(x) = 6x3 −11x2 + 6x −1

p(1) = 6(1)3 −11(1)2 + 6(1) −1 = 0

p(1) = 6(1)3 −11(1)2 + 6(1) −1 = 0
x −1 6x3 −11x2 + 6x −1 = 0

1
x −1 6x3 −11x2 + 6x −1 = 0

11116666 −−1−−11111 666 −−1−11

6666 −−−5−555 1111
666 −−−555 1111 0

6x2 − 5x +1

1

66xx33 −−111x2 + 6x −−11==((xx−−11))((66xx2 2−−55xx++1)1)

= (x −1)(2x −1)(3x −1)

x −1= 0 2x −1= 0 3x −1 = 0

1 x =xxx1===1111 xx=x=x12==121213 xx=x===1311313

2

x = 1, 1 , 1
 2 3

2

4x4 − 4x3 − 9x2 + x + 2 = 0

4x4 − 4x3 − 9x2 + x + 2 = 0

p(x) = 4x4 − 4x3 − 9x2 + x + 2

p(−1) = 4(−1)4 − 4(−1)3 − 9(−1)2 + (−1) + 2

= 4+ 4−9−1+ 2 = 0
= 4+ 4−9−1+ 2 = 0

2 4x4 − 4x3 − 9x2 + x + 2 = 0

p 1  = 4 1 4 − 4 1 3 − 9 1 2 +  1  + 2
2 2 2 2 2

= 4 1  − 4 1  − 9 1  +  1  + 2
16   8   4   2 

= 4 −4−9+1+2
16 8 4 2

= 4 − 8 − 36 + 8 + 32
16

=0

x +1 2x −1 4x4 − 4x3 − 9x2 + x + 2 = 0

x +1 2 4x4 − 4x3 −9x2 + x + 2
xxxxx+xxxxxx+++xx+1++++++111++111111111 ++4444444x114444xxx44xxx4xxxx444xx444−4444444−−−444−−−x−4−−−x−−4444+44x4+444xxx−44xxx31xxx3x313xx13333−333333−−−33443−−−−−−9−−−9x9x99999x99944x4xx99xxxx2xxx2−−22xx22222+222222++22+442++x++x+++xx+x+xxx+4xx−3x3xxxx++x+x−+1x++−−1++++++2+4224299222222−222xx 224++x 3xx−−++94922x 42
xx −+ 4xx+3 24−x94x−2 +4xx3+−29x2 + x + 2 =
12

44444x4444xxx444xx4x−xxx444xxx−44−4−−44−−−44444−−1−12−444−−−4−−111−2−112−1211−−411−122112−−−41141112241211441112224x444xxx444xx43x4444xx344x3443444x44xx34344444344−3444444343433344−44−4444−4334344444443−−44−4x−−−4944xx−−x−xx944949xx99x29xxx2x2944929xxxxxx22999−−2xx22−2x22222−x−−−xx22−22222xx−x−1−1221122−−−2−−−2−−2−−−−+22−6222−−−−+−−−−−+6462−24+2−6−−−−−2−−+−+−−6−6−−−−+4−6−−6−+−4−2−−6+−4x86++462−66−x−2−6x8xx−−82++x4+486x6664−2−x2−4xxx8846x42444x84644x264x46x82424343x4484x8−x66464−222x−xx8xx88−x4x+444+−x−x+x4+44−x−+−+4−4−−+−4−−++4+4+4++4+24242xx249−922444244422−22222x2x112−−−−−−−==−=−−==−−−−==−−−2−2−−==4−−==−−−−−==−−−−−−−−−−==−−−==−−−==−9−9−−9−1−1494==−−==14−==−−+14+6−−(96((9x−((−−(14(149−3331(4−((98388991489x(431(x41443xx6149xx468(282xx3(((83x(14(814xx84x43x((((3(3xxx834x884x83xxx83xx8+xx++x+x++x−−−+4+++++++++++++++1+1+1+x11+1+4149+4111−1−−1−212)1))−1)1−1)))111211−11)x11)(1)2(1−120(2−−110121))10011−(1)2)−)20)))2201−221122(2)211)))0)1−24(2(21)−022122(−210−0−11x2(22(==x2−==20−−0xx1012212222xx−xx−−22x−x2222x−x−xx+xx69x9−1x4x14xx=1=xx12((=(=−xx−x=3=−3−4868=−2−−xx−8xx=−4x−−==−x−−==4−−=−1−1−−−12111−−21211221+−−−+2x−1+21+22212212)2201+22)0111)0222)2011221101)022(4121(2221234(20)1(224212101())2002424)2420()04−)0(−4(02((4(4())()−((4((4114(4(4442x20(4x10414((x44x44(x(2x2(224(xx444x9xxx−x4−4xx4+xx+==x−x4+xxxx++x+−2x+xx22+x2x++x92=2+=2+x14+2−22−2(+(22−222222−23−2−−2282−22x−2−2x−2)−2)−1−2)12122−1−2)−)−(+(2262−(−−)2−626(()0)+60−)+x6(x66)x4()(66x2()xxx8(666xx(x46xx46x(64xx1(x(1xx−xx−6x−−−x−−x−4xx4x−)xx−x)−x+−−1−−−(−−x2−−021−2x2x−−−−222+42+2−−−2442422)2x−)24224)2424x)))4442)2)2)44))4)))−)−))=)4−)−))−))==)−))−(−)66x19x12142((2xx3)08−x−x(−−4(+2+24x44)11x)−)+))1(2201

22
14
2 −282 −−−133 −−222 0
2 44 −−626 −−−443 −002

4 44xx22 −−−666xx−−44 − 4 0
444xxx444 −−−444xxx333 −4−−x9992xxx−222+6++xxxx−+++4222 === (0(xx++11))xx == 11((44xx22 −−66xx−−44))

 22

4x4 − 4x3 − 9x2 + x + 2 == (xx++11)(22xxx=−−111)(44(4xxx++222−)(6xxx−−−224) )

 2

x x++1x1x=x+=+x0x+110xxx++1==++x+11=x0011+1==+0==1=100=00=0202xx−−2121xx2=x=−+x−2+021202x1−2x1x=2=x−1=x−2x−=01−0=x01−1−=01−0=11=0=1=0=0=000044=xx(2++x4x4x22+−x+4x==141+)x2−04x(04=2+2=x+1x4x042=+402+=x−xx4=20=+201++x=0)2=(2+240=0=x2=0+0x=0x−2−0)x24(2x−x=x=−+2x−04x022=−x2−xx)=20+−=2−x=0x2=0−2−x0=0=2
==xxx0−x===1=12xx12xxxxx−=−=x==+x=1=1x12=,−xx1,2x1x=x12=−xx112−==12x−xx====,1x1=,12x=−1−,−=x=12=12=0−,12−==12112212=12−12−1,,112,−x,12,1,2−221−−,12−,12−−,1112x1x−2,x,,1,1−12x,,1=21=2,−−=21212=12=x,,−−12−12,−,x212,=0−12,12x−,12212xx=2−1−2x2=,12=,=x2x−1,=x12x2x212,,==−12x=2−−=12x−xx1−2=11212−,x1x==−,1=2−−=12=12=−−12112120,,12−−,12122−41212,1xx−,,2x12+=12x=x,,2=−−2x=2=2=1212xx−02x,=x1=22=4x=2x−xx2=2=x12
x x==−xx−1x1==xx=x−x−x==11−==x=1x−−=−−−x=11111−+−111
x

2

2 (x +1)(2x −1) = 2x2 + x −1

2x2 + x −1 4x4 − 4x3 − 9x2 + x + 2 = 0

22xx22−−33xx−−22
2x2 + x −1 4x4 − 4x3 − 9x2 + x + 2

44xx44 ++22xx33 −−22xx22

− 6x33−−47xx222−+2xx + 2
−−66x33 − 33xx222 ++33xx

−−44xx222−−22xx++22
−−44xx222−−22xx++22

0

2x2 − 3x − 2

2

4x4 − 4x3 − 9x2 + x + 2 ==0(2x2 + x −1)(2x2 − 3x − 2)

= (x +1)(2x −1)(2x +1)(x − 2)

x +x1+2x=1x+20=−1x10=+=10x0=+10= 0 22x2x−x+21−x1=12−=0==100=02x0−1 = 02xx2−+x21+x=1+0=10=2x0+1 = 0 x −x 2−x=2−0=20=x0− 2 =

x +x1x=2==x−201−xx11=+=−110= 0x = −1 2x2−xx1+x−==1=212=0x1=20=0 1 x = 12x +−x12x====−00x−12=12 − 1 x = x−−12x x= =02x2 = 2
2 2 2 2

x = − 1,2, 1 ,− 1
 2 
2 

1. 1− 3x2 + 2x3 = 0
2. 2x4 −13x3 + 28x2 = 23x − 6