1729-2

Algebra Level 5

a b c + a b + b c + c a + a + b + c = 1 b c d + b c + c d + d b + b + c + d = 7 c d a + c d + d a + a c + c + d + a = 2 d a b + d a + a b + b d + d + a + b = 9 \begin{aligned} abc+ab+bc+ca+a+b+c&=&1\\ bcd+bc+cd+db+b+c+d&=&7 \\ cda+cd+da+ac+c+d+a&=&2\\ dab+da+ab+bd+d+a+b&=&9 \end{aligned} There exists a monic quartic polynomial with non-real roots a , b , c , d a,b,c,d . Compute its discriminant.

This is a sequel to this problem .


The answer is 49.

Aareyan Manzoor by Aareyan Manzoor , 18, USA

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2 solutions

Aareyan Manzoor
Dec 3, 2015

add one to each eqns to find: ( a + 1 ) ( b + 1 ) ( c + 1 ) = 2 (a+1)(b+1)(c+1)=2 ( b + 1 ) ( c + 1 ) ( d + 1 ) = 8 (b+1)(c+1)(d+1)=8 ( c + 1 ) ( d + 1 ) ( a + 1 ) = 3 (c+1)(d+1)(a+1)=3 ( d + 1 ) ( a + 1 ) ( b + 1 ) = 10 (d+1)(a+1)(b+1)=10 let A=a+1,B=b+1,C=c+1,D=d+1. then A B C = 2 , B C D = 8 , C D A = 3 , D A B = 10 ABC=2,BCD=8,CDA=3,DAB=10 multiply all to get ( A B C D ) 3 = 2 ∗ 8 ∗ 3 ∗ 10 = 480 (ABCD)^3=2*8*3*10=480 subtract each eqn simultaneously : { ( A − D ) B C = 2 − 8 = − 6 ( A − C ) B D = 10 − 8 = 2 ( A − B ) C D = 3 − 8 = − 5 ( B − C ) A D = 10 − 3 = 7 ( B − D ) A C = 2 − 3 = − 1 ( D − C ) A B = 10 − 2 = 8 \begin{cases} (A-D)BC=2-8=-6\\ (A-C)BD=10-8=2\\ (A-B)CD=3-8=-5\\ (B-C)AD=10-3=7\\ (B-D)AC=2-3=-1\\ (D-C)AB=10-2=8 \end{cases} multiply all these to get ( A − B ) ( A − C ) ( A − D ) ( B − C ) ( B − D ) ( D − C ) ( A B C D ) 3 = − 3360 (A-B)(A-C)(A-D)(B-C)(B-D)(D-C)(ABCD)^3=-3360 ( A − B ) ( A − C ) ( A − D ) ( B − C ) ( B − D ) ( D − C ) = − 7 (A-B)(A-C)(A-D)(B-C)(B-D)(D-C)=-7 Δ = ( ( a − b ) ( a − c ) ( a − d ) ( b − c ) ( b − d ) ( d − c ) ) 2 = 49 \Delta=((a-b)(a-c)(a-d)(b-c)(b-d)(d-c))^2=\boxed{49}

7 Helpful 0 Interesting 0 Brilliant 0 Confused
Lu Chee Ket
Dec 5, 2015

M y My H a r d Hard R o c k Rock M e t h o d : Method: 

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a0 =    -33.0108377293707-19.862417841431i  -33.0108377293707+19.862417841431i  0.0216754587411903
a1 =    -29.0487052561665-60.4626283912296i -29.0487052561665+60.4626283912296i 1.09741051233283
a2 =    7.19049975227237-40.995754279931i   7.19049975227237+40.995754279931i   3.61900049545527
a3 =    8.14323492023705-7.17629338954415i  8.14323492023705+7.17629338954415i  -4.28646984047409
a4 =    1   1   1

    -376488465.280024-837596977.285886i -376488465.280024+837596977.285886i 289.811131215241
    557018857.624071+1439827018.46873i  557018857.624071-1439827018.46873i  416.35775362945
    475161761.494311+1207613543.80348i  475161761.494311-1207613543.80348i  -228.330617505426
    -775721378.215706-2328555163.23013i -775721378.215706+2328555163.23013i -369.03560403146
    119675424.244892+533381271.923984i  119675424.244892-533381271.923984i  -39.1597786668287
    792949.012029856-14616861.1136552i  792949.012029856+14616861.1136552i  0.00260702297424232
    439148.879573792+52832.5665224288i  439148.879573792-52832.5665224288i  69.6454916639505
    -27112053.0573023-20480925.3338441i -27112053.0573023+20480925.3338441i -3483.57739539739
    49140101.1513267+41411478.4896755i  49140101.1513267-41411478.4896755i  -5630.27465324604
    36894281.3142384+30731609.5711795i  36894281.3142384-30731609.5711795i  2744.57152312857
    -73979969.871428-68618137.4405942i  -73979969.871428+68618137.4405942i  4928.74285596572
    -2213409.9916704-2284077.0311245i   -2213409.9916704+2284077.0311245i   -132.766659215398
    17825430.5353526+20236935.9594479i  17825430.5353526-20236935.9594479i  627.609294832842
    1499979.93435262-43919311.8499519i  1499979.93435262+43919311.8499519i  -20.498373689735
    362898.504861016+93490.2911534723i  362898.504861016-93490.2911534723i  -9115.12972203077
    -674629.371147559-208465.065309619i -674629.371147559+208465.065309619i 9575.26229511917
    208505.418533511+75463.6700144801i  208505.418533511-75463.6700144801i  -1676.4370670215
    128726.014361882+54508.6762329855i  128726.014361882-54508.6762329855i  903.171276236908
    -1939079.81392592+43866479.2834368i -1939079.81392592-43866479.2834368i -0.14711797421592
    49.0000004919712+7.32392072677612E-06i  49.0000004919712-7.32392072677612E-06i  48.9999999999996
    49.00000049 49.00000049 49

http://mathworld.wolfram.com/PolynomialDiscriminant.html

Better:

a = -0.0212830897077841

b = 1.60991176077924

c = -0.217026471766227

d = 2.91486764116886

Worse:

a = -1.48935845514611 ± \pm 0.847593707426474i

b = -2.30495588038962 ± \pm 2.2602498864706i

c = -1.39148676411689 ± \pm 0.67807496594118i

d = -2.95743382058443 ± \pm 3.3903748297059i

Answer: 49 \boxed{49}

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