Bairstow Method

"); popup.document.close(); } function tipclose(){ popup.window.close(); } function hlts(){ hlt=window.open('','','toolbar=no,width=350,height=250,location=top'); hlt.document.open(); hlt.document.writeln("highlights"); hlt.document.writeln("Advantages

Iteration scheme is simple to implement.
Iterations always converge for any continuous function f(x).

Disadvantages

Convergence process is slow.
Always one has to find the interval [a,b] which brackets the zero of f(x)= 0 more importantly f(a) * f(b) must be less than zero.

."); hlt.document.writeln(" "); hlt.document.close(); } function hltsclose(){ hlt.window.close(); }

1. Find the root of x³ - 11x²+ 32x - 22 = 0

Let p = 0.1 q = 0.1

[0] [1] [2] [3]

a[]

1.0 -11.0 32.0 -22.0

-0.1 1.11 -3.3010004 -3.3010004

-0.1 1.11
__________________________________________
b[]

1.0 -11.1 33.010002 -24.191

-0.1 1.1200001

-0.1
__________________________________________
c[]

1.0 -11.200001 34.030003

dp = -2.7770534 dq = 1.9070003

p = -2.6770535 q = 2.0070002

a[]

1.0 -11.0 32.0 -22.0

2.6770535 -22.280973 20.645508 -15.478039

-2.0070002 16.704155
_________________________________________
b[]

1.0 -8.322947 7.7120266 15.349663

2.6770535 -15.114358

-2.0070002
_________________________________________
c[]

1.0 -5.645893 -9.409331

dp = -1.2019181 dq = 0.9261256

p = -3.8789716 q = 2.9331257

a[]

1.0 -11.0 32.0 -22.0

3.8789716 -27.622267 5.6035914 -4.2372155

-2.9331257 20.886871
__________________________________________
b[]

1.0 -7.1210284 1.4446075 4.4904633

3.8789716 -12.575847

-2.9331257
__________________________________________
c[]

1.0 -3.2420568 -14.064364

dp = -0.35257483 dq = 0.3015399

p = -4.2315464 q = 3.2346656

a[]

1.0 -11.0 32.0 -22.0

4.2315464 -28.641026 0.52601856 -0.4020975

-3.2346656 21.893684
__________________________________________
b[]

1.0 -6.7684536 0.124308825 0.41970253

4.2315464 -10.735041

-3.2346656
__________________________________________
c[]

1.0 -2.5369072 -13.845398

dp = -0.036022577 dq = 0.032922886

p = -4.267569 q = 3.2675886

a[]

1.0 -11.0 32.0 -22.0

4.267569 -28.731113 0.0055390946 -0.004241169

-3.2675886 21.998816
_________________________________________
b[]

1.0 -6.732 0.001 0.004

4.267569 -10.518969

-3.2675886
_________________________________________
c[]

1.0 -2.4648619 -13.785259

dp = -3.8030953E-4 dq = 3.605403E-4

Final solution

p = -4.2679496 , q = 3.267949

Now

-p + Ö(p²- 4q)

root1 =
= 3.26

2

-p - Ö(p²- 4q)

root2 =
= 1

2

Now the deflated polynomial is

Q_n-2(x)* = x - 6.732 Þ root3 = 6.732

*(Q_n-2(x) is nothing but the first n-2 elements of b[ ]. )

So the roots of the equation are 3.26, 1.00 and 6.732

2. Find the root of x³ - x - 3 = 0

p = 0.1 q = 0.1

[0] [1] [2] [3]
a[]

1.0 -0.0 -1.0 -3.0

-0.1 0.010000001 0.109000005 0.109000005

-0.1 0.010000001
__________________________________________
b[]

1.0 -0.1 -1.09 -2.881

-0.1 0.020000001

-0.1
_________________________________________
c[]

1.0 -0.2 -1.1700001

dp = 25.82499 dq = 4.074998

p = 25.92499 q = 4.174998

a[]

1.0 -0.0 -1.0 -3.0

-25.92499 672.1051 -17290.156 -2784.432

-4.174998 108.23678
__________________________________________
b[]

1.0 -25.92499 666.9301 -17184.92

-25.92499 1344.2102

-4.174998
__________________________________________
c[]

1.0 -51.84998 2006.9653

dp = -12.900907 dq = -1.9816759

p = 13.024083 q = 2.193322

a[]

1.0 -0.0 -1.0 -3.0

-13.024083 169.62674 -2167.6426 -365.04205

-2.193322 28.566008
__________________________________________
b[]

1.0 -13.024083 166.43341 -2142.0767

-13.024083 339.25348

-2.193322
__________________________________________
c[]

1.0 -26.048166 503.49356

dp = -6.4232802 dq = -0.8812569

p = 6.600803 q = 1.3120651

a[]

1.0 -0.0 -1.0 -3.0

-6.600803 43.5706 -272.33945 -54.133884

-1.3120651 8.660684
__________________________________________
b[]

1.0 -6.600803 41.258533 -266.67877

-6.600803 87.1412

-1.3120651
__________________________________________
c[]

1.0 -13.201606 127.08767

dp = -3.142904 dq = -0.23284553

p = 3.4578989 q = 1.0792196

a[]

1.0 -0.0 -1.0 -3.0

-3.4578989 11.957065 -34.15659 -10.660363

-1.0792196 3.7318323
__________________________________________
b[]

1.0 -3.4578989 9.877845 -33.42476

-3.4578989 23.91413

-1.0792196
__________________________________________
c[]

1.0 -6.9157977 32.712753

dp = -1.3959081 dq = 0.22402748

p = 2.0619907 q = 1.3032471

a[]

1.0 -0.0 -1.0 -3.0

-2.0619907 4.251806 -4.01791 -2.5394535

-1.3032471 2.6872835
__________________________________________
b[]

1.0 -2.0619907 1.9485587 -4.3306265

-2.0619907 8.503612

-1.3032471
__________________________________________
c[]

1.0 -4.1239815 9.148923

dp = -0.37781653 dq = 0.3904503

p = 1.6841742 q = 1.6936975

a[]

1.0 -0.0 -1.0 -3.0

-1.6841742 2.8364427 -0.24040787 -0.24176727

-1.6936975 2.8524816
__________________________________________
b[]

1.0 -1.6841742 0.14274526 -0.38792634

-1.6841742 5.6728854

-1.6936975
__________________________________________
c[]

1.0 -3.3683484 4.1219335

dp = -0.012609564 dq = 0.100271836

p = 1.6715646 q = 1.7939693

a[]

1.0 -0.0 -1.0 -3.0

-1.6715646 2.7941282 -2.656216E-4 -2.8507246E-4

-1.7939693 2.9987354
__________________________________________
b[]

1.0 -1.6715646 1.5890598E-4 -0.0015301704

-1.6715646 5.5882564

-1.7939693
__________________________________________
c[]

1.0 -3.3431292 3.794446

dp = 1.3531519E-4 dq = 6.112821E-4

Final solution
p = 1.6716999 , q = 1.7945806

Now

-p + Ö(p²- 4q)

root1 =
= -0.835+ 1.046 i

2

-p - Ö(p²- 4q)

root2 =
= -0.835- 1.046 i

2

Q_n-2(x)* = x - 1.6716 Þroot3 = 1.6716

so the roots of the equation are -0.835+ 1.046 i, -0.835 - 1.046 i and 1.6716

3. Find the root of x⁴+ x³+ 2x² + x + 1 = 0

p = 0.1 q = 0.1

[0] [1] [2] [3] [4]
a[]

1.0 1.0 2.0 1.0 1.0

-0.1 -0.089999996 -0.181 -0.072900005 -0.072900005

-0.1 -0.089999996 -0.181
__________________________________________
b[]

1.0 0.9 1.81 0.72900003 0.7461

-0.1 -0.08 -0.16299999

-0.1 -0.08
__________________________________________
c[]

1.0 0.79999995 1.6299999 0.48600006

dp = 0.20741068 dq = 0.4886508

p = 0.3074107 q = 0.5886508

a[]

1.0 1.0 2.0 1.0 1.0

-0.3074107 -0.21290934 -0.3684132 -0.06882727 -0.13179512

-0.5886508 -0.40769324 -0.7054626
__________________________________________
b[]

1.0 0.6925893 1.1984398 0.22389355 0.22571015

-0.3074107 -0.11840802 -0.15105578

-0.5886508 -0.2267357
__________________________________________
c[]

1.0 0.3851786 0.49138105 -0.15389793

dp = 0.05963816 dq = 0.5051903

p = 0.36704886 q = 1.0938411

a[]

1.0 1.0 2.0 1.0 1.0

-0.36704886 -0.23232399 -0.24733034 -0.022141028 -0.06598241

-1.0938411 -0.69234794 -0.7370683
__________________________________________
b[]

1.0 0.63295114 0.6738349 0.06032175 0.24079067

-0.36704886 -0.09759913 0.18998644

-1.0938411 -0.29085484
__________________________________________
c[]

1.0 0.26590228 -0.5176053 -0.040546656

dp = -0.32316893 dq = -0.4022237

p = 0.043879926 q = 0.69161737

a[]

1.0 1.0 2.0 1.0 1.0

-0.043879926 -0.04195448 -0.055570774 -0.012425039 -0.19583835

-0.69161737 -0.66126925 -0.8758837
__________________________________________
b[]

1.0 0.9561201 1.2664281 0.28315997 0.11169124

-0.043879926 -0.04002903 -0.023466181

-0.69161737 -0.6309211
__________________________________________
c[]

1.0 0.91224015 0.5347817 -0.37122732

dp = 0.05610686 dq = 0.2775092

p = 0.09998679 q = 0.9691266

a[]

1.0 1.0 2.0 1.0 1.0

-0.09998679 -0.08998943 -0.09407597 -0.0033692857 -0.032656956

-0.9691266 -0.8722267 -0.9118357
__________________________________________
b[]

1.0 0.9000132 0.940884 0.033697307 0.08479506

-0.09998679 -0.07999208 0.010822034

-0.9691266 -0.77532685
__________________________________________
c[]

1.0 0.8000264 -0.108234644 -0.7308075

dp = -0.11468163 dq = 0.026605101

p = -0.01469484 q = 0.9957317

a[]

1.0 1.0 2.0 1.0 1.0

0.01469484 0.014910778 0.014976672 6.778441E-5 -0.004593115

-0.9957317 -1.0103638 -1.0148288
__________________________________________
b[]

1.0 1.0146948 1.019179 0.0046128035 -0.014760971

0.01469484 0.015126715 5.668385E-4

-0.9957317 -1.0249959
__________________________________________
c[]

1.0 1.0293896 0.03857398 -1.0198163

dp = 0.014557164 dq = 0.0039356095

p = -1.3767555E-4 q = 0.99966735

a[]

1.0 1.0 2.0 1.0 1.0

1.3767555E-4 1.3769451E-4 1.3774031E-4 4.5806463E-8 -3.3260245E-4

-0.99966735 -0.999805 -1.0001376
__________________________________________
b[]

1.0 1.0001377 1.0004704 3.3271313E-4 -1.3756752E-4

1.3767555E-4 1.3771346E-4 1.2951692E-7

-0.99966735 -0.9999426
__________________________________________
c[]

1.0 1.0002754 9.407401E-4 -0.99960977

dp = 1.3788824E-4 dq = 3.3249185E-4

Final solution
p = 2.126908E-7 , q = 0.9999998

\ root1= i and root2 = -i

Q_n-2(x) = x² + x +1 \ root3 = -0.499 + 0.866 i and root4 = -0.499 - 0.866 i

so the roots of the given equation are i, -i, -0.499 + 0.866 i and -0.499 - 0.866 i

4. Find the root of x⁶- x⁴- x³ - 1 = 0

p = 0.1 q = 0.1

[0] [1] [2] [3] [4] [5] [6]
a[]

1.0 0.0 -1.0 -1.0 0.0 0.0 -1.0

-0.1 0.0100 0.1090 0.0881 -0.0197 -0.0068 -0.0068

-0.1 0.010000001 0.109000005 0.0881 -0.019710002
__________________________________________
b[]

1.0 -0.1 -1.09 -0.881 0.19710001 0.06839 -1.026549

-0.1 0.020000001 0.117000006 0.0744 -0.038850006

-0.1 0.020000001 0.117000006 0.0744
__________________________________________
c[]

1.0 -0.2 -1.1700001 -0.744 0.38850003 0.103939995

dp = -4.1559186 dq = -2.262049

p = -4.0559187 q = -2.162049

a[]

1.0 0.0 -1.0 -1.0 0.0 0.0 -1.0

4.055 16.450 71.43 321.24 1457.3904 6605.605 3521.1853

2.162049 8.769095 38.07914 171.24307 776.8769
__________________________________________
b[]

1.0 4.0559187 17.612524 79.20406 359.32437 1628.6334 7381.482

4.0559187 32.90095 213.64763 1258.916 7025.368

2.162049 17.53819 113.88706 671.0781
__________________________________________
c[]

1.0 8.111837 52.67552 310.38986 1732.1274 9325.08

dp = 0.8666872 dq = 0.41051808

p = -3.1892314 q = -1.751531

a[]

1.0 0.0 -1.0 -1.0 0.0 0.0 -1.0 23541.254

3.189 10.171 34.83 125.723 461.97 1693.5531 930.1021

1.751531 5.5860376 19.131496 69.047356 253.7174
__________________________________________
b[]

1.0 3.1892314 10.922728 39.421143 144.85464 531.02234 1946.2705

3.1892314 20.342394 105.29774 497.1723 2232.005

1.751531 11.172075 57.82969 273.04782
__________________________________________
c[]

1.0 6.378463 33.01665 155.89095 699.8567 3036.0752

dp = 0.6872543 dq = 0.3210111

p = -2.501977 q = -1.4305199

a[]
1.0 0.0 -1.0 -1.0 0.0 0.0 -1.0

2.501 6.259 16.739 48.33 144.87 431.62 246.78

1.4305199 3.579128 9.570763 27.635323 82.83411
__________________________________________
b[]

1.0 2.501 6.690 19.318 57.904 172.51 513.4552

2.501977 12.519777 51.64257 195.45244 707.76996

1.4305199 7.158256 29.52694 111.751076
__________________________________________
c[]

1.0 5.003954 20.640705 78.1192 282.88428 992.0331

dp = 0.5430311 dq = 0.24190012

p = -1.9589459 q = -1.1886199

a[]

1.0 0.0 -1.0 -1.0 0.0 0.0 -1.0 1005.83093

1.9589459 3.8374689 7.88689 18.052336 44.73 109.09 66.19

1.1886199 2.3284419 4.785489 10.9535265 27.145493
__________________________________________
b[]

1.0 1.9589459 4.0260887 9.215332 22.837826 55.69159 135.24231

1.9589459 7.6749377 25.25012 76.63854 224.8816

1.1886199 4.6568837 15.320889 46.501587
__________________________________________
c[]

1.0 3.9178917 12.889647 39.122337 114.79726 327.07477

dp = 0.4303523 dq = 0.1607352

p = -1.5285935 q = -1.0278846

a[]

1.0 0.0 -1.0 -1.0 0.0 0.0 -1.0

1.528 2.336 3.6143 6.398 13.49 27.204 18.29

1.0278846 1.5712178 2.4304156 4.3022633 9.074598
__________________________________________
b[]

1.0 1.5285935 2.3644829 4.1855507 8.828422 17.797333 35.279488

1.5285935 4.6731963 12.328969 30.047493 72.098236

1.0278846 3.1424356 8.290469 20.20508
__________________________________________
c[]

1.0 3.057187 8.065564 19.656954 47.166386 110.10065

dp = 0.3557414 dq = 0.051803105

p = -1.1728522 q = -0.9760815

a[]

1.0 0.0 -1.0 -1.0 0.0 0.0 -1.0

1.1728522 1.3755822 1.5853018 2.0291526 3.9272795 6.5867367 5.481673

0.9760815 1.1447992 1.3193339 1.6887195 3.2683957
__________________________________________
b[]

1.0 1.1728522 1.3516637 1.730101 3.3484864 5.615999 8.855132

1.1728522 2.7511644 5.95681 11.700971 23.46512

0.9760815 2.2895985 4.9574294 9.737885
__________________________________________
c[]

1.0 2.3457043 5.0789094 9.976509 20.006887 38.819004

dp = 0.34792018 dq = -0.13479656

p = -0.824932 q = -1.110878

a[]

1.0 0.0 -1.0 -1.0 0.0 0.0 -1.0

0.824932 0.6805128 0.65284353 0.46958616 1.1126062 1.4394773 1.9384431

1.110878 0.91639876 0.8791386 0.63235873 1.4982686
__________________________________________
b[]

1.0 0.824932 0.7913908 0.5692423 1.3487247 1.7449648 1.9377459

0.824932 1.3610256 2.6919959 4.202233 7.5696416

1.110878 1.8327975 3.6251218 5.658852
__________________________________________
c[]

1.0 1.649864 3.2632942 5.0940356 9.17608 14.973459

dp = 0.36523032 dq = -0.31535277

p = -0.45970166 q = -1.4262308

a[]

1.0 0.0 -1.0 -1.0 0.0 0.0 -1.0

0.45970166 0.21132562 0.29308575 -0.023570552 0.4071725 0.15356083 0.4764246

1.4262308 0.65564066 0.9093026 -0.07312796 1.2632583
__________________________________________
b[]

1.0 0.459 0.6375 -0.05127 0.8857 0.3340 0.4168

0.45970166 0.42265123 1.1430198 1.1046758 2.5452037

1.4262308 1.3112813 3.5462348 3.4272718
__________________________________________
c[]

1.0 0.9194033 2.4864383 2.4030275 5.5366426 6.30652

dp = 0.05200817 dq = 0.019181592

p = -0.40769348 q = -1.4070492

a[]

1.0 0.0 -1.0 -1.0 0.0 0.0 -1.0

0.407 0.166 0.2337 -0.0785 0.2968 0.01054 0.036

1.4070492 0.57364476 0.80660945 -0.27105337 1.0244325
__________________________________________
b[]

1.0 0.4076 0.573 -0.1926 0.728 0.0257 0.0349

0.4076 0.3324 0.9428 0.7736 1.9389

1.4070492 1.1472895 3.2541394 2.669931

__________________________________________
c[]

1.0 0.81538695 2.3127403 1.8975391 4.755825 4.6346264

dp = 0.0040574386 dq = 0.003415048

p = -0.40363604 q = -1.4036342

a[]

1.0 0.0 -1.0 -1.0 0.0 0.0 -1.0

0.403 0.1629 0.228 -0.0826 0.287 8.8E-5 3.06E-4

1.4036342 0.56655735 0.79523766 -0.28740832 1.0002145
__________________________________________
b[]

1.0 0.4036 0.566 -0.204 0.712 2.18E-4 3.02E-4

0.40363604 0.3258441 0.9267623 0.748792 1.8907013

1.4036342 1.1331147 3.2227926 2.6039054
__________________________________________
c[]

1.0 0.8072721 2.2960346 1.8551168 4.6841736 4.494825

dp = 3.3909968E-5 dq = 3.2052867E-5

Final solution
p = -0.40360212 , q = -1.4036021

\ root1= -1 and root2 = 1.404

Q_n-2(x) =1.0x⁴ + 0.4036x³ + 0.566x² - 0.204x + 0.712

The other four roots are found by repeating the above procedure using Q_n-2(x).

5. Find the root of x³- 3.7x²+ 6.2x - 4.069 = 0

p = 0.1 q = 0.1

[0] [1] [2] [3]
a[]

1.0 -3.7 6.2 -4.069

-0.1 0.38 -0.648 -0.648

-0.1 0.38
__________________________________________
b[]

1.0 -3.8 6.48 -4.337

-0.1 0.39

-0.1
__________________________________________
c[]

1.0 -3.8999999 6.77

dp = -1.4031502 dq = 1.0077145

p = -1.3031502 q = 1.1077145

a[]

1.0 -3.7 6.2 -4.069

1.3031502 -3.1234553 2.5656812 -2.1809015

-1.1077145 2.6550255
__________________________________________
b[]

1.0 -2.3968499 1.96883 1.1517069

1.3031502 -1.425255

-1.1077145
__________________________________________
c[]

1.0 -1.0936997 -0.5641395

dp = -0.8862656 dq = 0.9995217

p = -2.1894157 q = 2.1072361

a[]

1.0 -3.7 6.2 -4.069

2.1894157 -3.307297 1.7197131 -1.6551638

-2.1072361 3.183158
__________________________________________
b[]

1.0 -1.5105844 0.7854667 0.83387136

2.1894157 1.486244

-2.1072361
__________________________________________
c[]

1.0 0.67883134 0.16447449

dp = -0.2779357 dq = 0.97413814

p = -2.4673514 q = 3.0813742

a[]

1.0 -3.7 6.2 -4.069

2.4673514 -3.0413773 0.19059879 -0.23803103

-3.0813742 3.7982516
__________________________________________
b[]

1.0 -1.2326486 0.077248335 -0.08014941

2.4673514 3.0464458

-3.0813742

__________________________________________
c[]

1.0 1.2347028 0.042320013

dp = 0.11255993 dq = -0.06172974

p = -2.3547914 q = 3.0196445

a[]

1.0 -3.7 6.2 -4.069

2.3547914 -3.1676857 0.02983418 -0.038257577

-3.0196445 4.062052
__________________________________________
b[]

1.0 -1.3452086 0.012669563 0.022886276

2.3547914 2.3773568

-3.0196445
__________________________________________
c[]

1.0 1.0095828 -0.62961817

dp = -0.0060758535 dq = 0.01880364

p = -2.3608673 q = 3.038448

a[]

1.0 -3.7 6.2 -4.069

2.3608673 -3.1615148 8.7245564E-5 -1.1228548E-4

-3.038448 4.0688853
__________________________________________
b[]

1.0 -1.339 3.695488E-5 -2.7179718E-5

2.3608673 2.4121795

-3.038448
__________________________________________
c[]

1.0 1.0217345 -0.6262317

dp = 3.8880014E-5 dq = -2.770174E-6

Final solution
p = -2.3608284 , q = 3.0384452

\ root1= 1.18 + 1.283i and root2 = 1.18 - 1.283i

Q_n-2(x) = x - 1.339 Þ root3 = 1.339

so the three roots of the given equation are 1.18 + 1.283i, 1.18 - 1.283i and 1.339

6. Find the root of x⁴- x - 10 = 0

p = 0.1 q = 0.1

[0] [1] [2] [3] [4]
a[]

1.0 -0.0 -0.0 -1.0 -10.0

-0.1 0.010000001 0.009000001 0.0981 0.0981

-0.1 0.010000001 0.009000001
__________________________________________
b[]

1.0 -0.1 -0.09 -0.981 -9.8929

-0.1 0.020000001 0.017

-0.1 0.020000001
__________________________________________
c[]

1.0 -0.2 -0.17 -0.944

dp = -49.912125 dq = 47.330303

p = -49.812126 q = 47.4303

a[]

1.0 -0.0 -0.0 -1.0 -10.0

49.812126 2481.2478 121233.62 5921168.0 -5638040.5

-47.4303 -2362.6042 -115436.695
__________________________________________
b[]

1.0 49.812126 2433.8174 118870.016 5805721.5

49.812126 4962.4956 366063.47

-47.4303 -4725.2085
__________________________________________
c[]

1.0 99.62425 7348.883 480208.28

dp = 16.391087 dq = -15.92156

p = -33.42104 q = 31.508741

a[]

1.0 -0.0 -0.0 -1.0 -10.0

33.42104 1116.966 36277.105 1177191.0 -1109834.0

-31.508741 -1053.0549 -34201.39
__________________________________________
b[]

1.0 33.42104 1085.4572 35223.05 1142979.6

33.42104 2233.932 109884.38

-31.508741 -2106.1099
__________________________________________
c[]

1.0 66.84208 3287.8804 143001.33

dp = 10.92892 dq = -10.620994

p = -22.492119 q = 20.887749

a[]

1.0 -0.0 -0.0 -1.0 -10.0

22.492119 505.89542 10908.85 234773.64 -218027.16

-20.887749 -469.80972 -10130.718
__________________________________________
b[]

1.0 22.492119 485.00766 10438.04 224632.92

22.492119 1011.79083 33196.36

-20.887749 -939.61945
__________________________________________
c[]

1.0 44.984238 1475.9108 42694.777

dp = 7.288478 dq = -7.0936403

p = -15.203641 q = 13.794108

a[]

1.0 -0.0 -0.0 -1.0 -10.0

15.203641 231.1507 3304.6113 47038.402 -42677.46

-13.794108 -209.72067 -2998.2402
__________________________________________
b[]

1.0 15.203641 217.35658 3093.8906 44030.164

15.203641 462.3014 10123.555

-13.794108 -419.44135
__________________________________________
c[]

1.0 30.407282 665.86383 12798.004

dp = 4.8636374 dq = -4.7564197

p = -10.340004 q = 9.037689

a[]

1.0 -0.0 -0.0 -1.0 -10.0

10.340004 106.91568 1012.05884 9488.081 -8293.065

-9.037689 -93.449745 -884.5909
_____________________________________
b[]

1.0 10.340004 97.87799 917.6091 8593.49

10.340004 213.83136 3129.6262

-9.037689 -186.89949
__________________________________________
c[]

1.0 20.680008 302.67166 3860.336

dp = 3.252231 dq = -3.2277126

p = -7.0877733 q = 5.8099766

a[]

1.0 -0.0 -0.0 -1.0 -10.0

7.0877733 50.23653 314.8853 1932.8748 -1584.4126

-5.8099766 -41.1798 -258.11722
__________________________________________
b[]

1.0 7.0877733 44.42655 272.7055 1664.7576

7.0877733 100.47306 985.83575

-5.8099766 -82.3596
__________________________________________
c[]

1.0 14.175547 139.08963 1176.1816

dp = 2.1918354 dq = -2.2684178

p = -4.895938 q = 3.5415587

a[]

1.0 -0.0 -0.0 -1.0 -10.0

4.895938 23.970207 100.017395 399.89114 -289.26794

-3.5415587 -17.339252 -72.34926
__________________________________________
b[]

1.0 4.895938 20.428648 81.67815 317.54187

4.895938 47.940414 317.39145

-3.5415587 -34.678505
______________________________________
c[]

1.0 9.791876 64.82751 364.39108

dp = 1.5238351 dq = -1.7471913

p = -3.3721027 q = 1.7943674

a[]

1.0 -0.0 -0.0 -1.0 -10.0

3.3721027 11.371077 32.293648 85.121506 -45.29496

-1.7943674 -6.0507913 -17.184134
__________________________________________
b[]

1.0 3.3721027 9.576709 25.242857 57.93737

3.3721027 22.742153 102.93174

-1.7943674 -12.101583
__________________________________________
c[]

1.0 6.7442055 30.524496 116.073006

dp = 1.1899198 dq = -1.6427208

p = -2.1821828 q = 0.15164661

a[]

1.0 -0.0 -0.0 -1.0 -10.0

2.1821828 4.761922 10.060463 19.049458 -1.3238056

-0.15164661 -0.33092064 -0.6991326
__________________________________________
b[]

1.0 2.1821828 4.6102753 8.729543 8.350325

2.1821828 9.523844 30.51231

-0.15164661 -0.6618413
__________________________________________
c[]

1.0 4.3643656 13.982472 38.58001

dp = 1.3125122 dq = -2.2048159

p = -0.8696706 q = -2.0531693

a[]

1.0 -0.0 -0.0 -1.0 -10.0 7.262032

0.8696706 0.75632703 2.4433365 2.8080947 6.629514

2.0531693 1.785581 5.7683716
__________________________________________
b[]

1.0 0.8696706 2.8094964 3.2289176 -1.4235334

0.8696706 1.5126541 5.544428

2.0531693 3.571162
__________________________________________
c[]

1.0 1.7393413 6.3753195 12.344508

dp = 0.9302859 dq = -1.5534341

p = 0.06061524 q = -3.6066034

a[]

1.0 -0.0 -0.0 -1.0 -10.0

-0.06061524 0.0036742075 -0.21883786 0.08713156 -5.184323

3.6066034 -0.21861513 13.02084
__________________________________________
b[]

1.0 -0.06061524 3.6102777 -1.437453 3.1079712

-0.06061524 0.007348415 -0.4378984

3.6066034 -0.43723026
__________________________________________
c[]

1.0 -0.12123048 7.2242293 -2.3125818

dp = -0.19214778 dq = 0.4069385

p = -0.13153253 q = -3.1996648

a[]

1.0 -0.0 -0.0 -1.0 -10.0

0.13153253 0.017300807 0.42313564 -0.020519646 -0.4991616

3.1996648 0.42086002 10.293212
__________________________________________
b[]

1.0 0.13153253 3.2169657 -0.15600434 0.27269268

0.13153253 0.034601614 0.8485469

3.1996648 0.84172004
__________________________________________
c[]

1.0 0.26306507 6.451232 1.5342627

dp = -0.026185522 dq = 0.04913065

p = -0.15771806 q = -3.1505342

a[]

1.0 -0.0 -0.0 -1.0 -10.0

0.15771806 0.024874987 0.5008194 -3.6030175E-4 -0.007197292

3.1505342 0.49689615 10.004234

__________________________________________
b[]

1.0 0.157 3.17 -0.00 0.00

0.15771806 0.049749974 1.005562

3.1505342 0.9937923
__________________________________________
c[]

1.0 0.31543612 6.3756933 1.9970698

dp = -3.9448982E-4 dq = 7.3130085E-4

Final solution
p = -0.15811256 , q = -3.149803

\ root1= -1.697 and root2 = 1.856

Q_n-2(x) = x² + 0.157x + 3.17 \ root3 = -0.079 + 1.78 i and root4 = -0.079 - 1.78 i

so the roots of the given equation are -1.697, 1.856, -0.079 + 1.78 i and -0.079 - 1.78 i

Problems to Work-Out:

7. Find the root of x⁴- x - 4 = 0

8. Find the root of 2x³ - 3x²+ 2x - 3 = 0

9. Find the root of x³- 5x²+ 4x - 3 = 0

10. Find the root of x³- x²- x + 1 = 0

11. Find the root of 9x⁴+ 30x³+ 34x²+ 30x + 25 = 0

12. Find the root of x⁵- 2x⁴+4x³-x²-7x+5= 0