document.write( "Question 1175454: Find a cubic polynomial with integer coefficients that has \"matrix%282%2C1%2C%22%22%2C+2%5E%281%2F3%29%2B4%5E%281%2F3%29%29++\" as a root.\r
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\n" ); document.write( "\n" ); document.write( "I've been working on this question for some time, but I haven't made any headway. Can somebody write a solution to this? \n" ); document.write( "
Algebra.Com's Answer #801086 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "See the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Prove that the number (cube root of 2 PLUS cube root of 4) is irrational\r
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\n" ); document.write( "\n" ); document.write( "This problem is  ADVANCED  and the solution is intended for  ADVANCED  STUDENTS.\r
\n" ); document.write( "\n" ); document.write( "Actually,  it is a  Math  Circle level problem.   Below is the copy of this elegant solution.\r
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Problem

(a)   Find a cubic polynomial with integer coefficients that has   \"root%283%2C2%29\" + \"root%283%2C4%29\"   as a root.
\n" ); document.write( "(b)   Prove that the number   \"root%283%2C2%29\" + \"root%283%2C4%29\"   is irrational.\r
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\n" ); document.write( "\n" ); document.write( "Part (a)\r
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document.write( "Let r = \"root%283%2C2%29\" + \"root%283%2C4%29\".\r\n" );
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document.write( "Then,  since  (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + 3ab*(a+b) + b^3\r\n" );
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document.write( "    r^3 = \"%28root%283%2C2%29%29%5E3\" + \"3%2Aroot%283%2C2%29%2Aroot%283%2C4%29\".\"%28root%283%2C2%29+%2B+root%283%2C4%29%29\" + \"%28root%283%2C4%29%29%5E3\" = \r\n" );
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document.write( "        = 2 + \"3%2Aroot%283%2C8%29\".\"%28root%283%2C2%29%2Broot%283%2C4%29%29\" + 4 = 2 + 3*2*r + 4 = 6 + 6r.\r\n" );
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document.write( "It means that  r = \"root%283%2C2%29\" + \"root%283%2C4%29\"  is the root to this equation\r\n" );
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document.write( "    r^3 - 6r - 6 = 0.      (*)\r\n" );
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document.write( "ANSWER.   \"root%283%2C2%29\" + \"root%283%2C4%29\"  is the root of the cubic polynomial  x^3 - 6x - 6.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Part (a) is solved.\r
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\n" ); document.write( "\n" ); document.write( "Part (b)\r
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document.write( "In part (a), I proved that the number  r = \"root%283%2C2%29\" + \"root%283%2C4%29\"  is the root of the cubic equation\r\n" );
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document.write( "    x^3 - 6x - 6 = 0.     (**)\r\n" );
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document.write( "Therefore, due to the Rational root theorem, if the number \"r\" is rational, it must divide the constant term of 6, \r\n" );
document.write( "i.e. r must be one of the numbers +/-1, +/-2, +/-3, +/-6.\r\n" );
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document.write( "But it is easy to check that no one of these divisors of 6 IS NOT THE ROOT to equation (**).\r\n" );
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document.write( "Indeed, for these values of x, the values of the polynomial  f(x) = x^3 - 6x -6  are given in the Table\r\n" );
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document.write( "    x       -1      1     -2      2     -3      3     -6     6\r\n" );
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document.write( "    f(x)    -1    -11     -2    -10    -15      3    -186   174\r\n" );
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document.write( "and no one of these values of the polynomial  f(x)  is equal to  0  (zero).\r\n" );
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\n" ); document.write( "\n" ); document.write( "I am reading the Edwins' comments about my work at this forum,  and can not understand what he wants to say.\r
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\n" ); document.write( "\n" ); document.write( "Edwin,  don't you think,  that it would be better if you take them back  (or take them off)  and will not comment my posts ?\r
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\n" ); document.write( "\n" ); document.write( "I have no objections when somebody  (colleagues tutors)  point me to my error (which happens not so often),\r
\n" ); document.write( "\n" ); document.write( "moreover,  I always thankful for it . . . \r
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\n" ); document.write( "\n" ); document.write( "But I always feel myself  UNCOMFORTABLE,  when I see comments,  that I did not deserve  (or that are nonsensical).\r
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\n" ); document.write( "\n" ); document.write( "                About  @ikleyn job at this forum the visitors and the tutors should know one thing:\r
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\n" ); document.write( "\n" ); document.write( "                        WHAT  @ikleyn  DOES  at this forum  IS  ALWAYS  RIGHT .\r
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