document.write( "Question 1039987: x+y+z=12;xy+yz+zx=44;x^3+y^3+z^3=288. Find value of x,y and z. \n" ); document.write( "

Algebra.Com's Answer #654842 by ikleyn(52786) $\"\"$ $\"About$

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\n" ); document.write( "x+y+z=12; xy+yz+zx=44; x^3+y^3+z^3=288. Find value of x,y and z.
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\n" ); document.write( "\n" ); document.write( " $\"system%28x%2By%2Bz=12%2C%0D%0Axy%2Byz%2Bzx=44%2C%0D%0Ax%5E3%2By%5E3%2Bz%5E3=288%29\"$ (1)\r
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document.write( "The previous tutor found the solution by the method of \"trial and error\".\r\n" );
document.write( "The question remains still open if it can be solved on the more solid algebraic base.\r\n" );
document.write( "The answer is \"Yes\", and I will show you \"How\".\r\n" );
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document.write( "With one apology: since the problem is slightly higher than the traditional school math, the solution is, correspondingly, \r\n" );
document.write( "slightly higher (but still understandable). \r\n" );
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\n" ); document.write( "\n" ); document.write( "Part 1. Motivation\r
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document.write( "We are given the numerical values for the functions ,  and . \r\n" );
document.write( "A remarkable fact is that the function  can be explicitly expressed via ,  and . (I will do it later).\r\n" );
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document.write( "So, knowing the numerical values of ,  and , we can find the value of .\r\n" );
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document.write( "Next, knowing the numerical values of ,  and , we can consider the polynomial of the variable \"u\"\r\n" );
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document.write( "P(u) = .   (2)\r\n" );
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document.write( "This polynomial is nothing else as \r\n" );
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document.write( "P(u) = (u-x)*(u-y)*(u-z).     (3)\r\n" );
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document.write( "In other words, the polynomial (2) is factorable into (3).\r\n" );
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document.write( "Now, instead of solving the system (1), we can solve the polynomial equation \r\n" );
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document.write( "P(u) = 0      (4)\r\n" );
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document.write( "for u. If we solve it (and when we solve it), its roots u = x, u = y and u = z will be the solution of the system (1).\r\n" );
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document.write( "Why this way is better than solving (1) directly?\r\n" );
document.write( "Well, for example, you can (try) to solve the polynomial equation (4) graphically. \r\n" );
document.write( "Or apply other methods specific for polynomial equations. \r\n" );
document.write( "You will see it later.\r\n" );
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document.write( "Now I will implement this methodology.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Part 2. Calculation of $\"xyz\"$ via $\"x%2By%2Bz\"$ , $\"xy%2Byz%2Bzx\"$ and $\"x%5E3%2By%5E3%2Bz%5E3\"$ \r
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document.write( "We will do it step by step:\r\n" );
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document.write( "1.   = \r\n" );
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document.write( "2.    = \r\n" );
document.write( "   = x(xy+xz) + y(xy+yz) + z(xz+yz) = \r\n" );
document.write( "   = x(xy+xz+yz-yz) + y(xy+yz+xz-xz) + z(xz+yz+xy-xy)\r\n" );
document.write( "   = x(xy+xz+yz) + y(xy+xz+yz) + z(xy+xz+yz) - x(yz) - y(xz) - z(xy)\r\n" );
document.write( "   = (x+y+z)(xy+xz+yz) - 3xyz\r\n" );
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document.write( "3.  Therefore, \r\n" );
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document.write( "      = ,   or\r\n" );
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document.write( "      = ,   or\r\n" );
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document.write( "      = ,   or\r\n" );
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document.write( "     xyz = . \r\n" );
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document.write( "4.   Thus\r\n" );
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document.write( "     xyz =  =  = 48.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Part 3. Working with the polynomial\r
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document.write( "So, our polynomial (2) is\r\n" );
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document.write( "    P(u) = ,\r\n" );
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document.write( "and we need solve this polynomial equation (4)\r\n" );
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document.write( "     = .\r\n" );
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document.write( "First, let's do it graphically.

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\n" ); document.write( "\n" ); document.write( "Figure. Plot P(u) = $\"u%5E3+-+12u%5E2+%2B+44u+-+48\"$ \r
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Do you see the roots? Of course, they are u=2, u=4 and u=6.\r\n" );
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document.write( "And you can check it manually substituting these values into the polynomial.\r\n" );
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document.write( "Or you can apply the rational roots theorem.\r\n" );
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document.write( "All the roots are among the integer divisors of the number 48, and you have only finite number of options to check.\r\n" );
document.write( "It is your other method to find the roots.\r\n" );
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\n" ); document.write( "\n" ); document.write( "So, the original problem is solved algebraically.
\n" ); document.write( "The solution is x=2, y=3, z=6 and all permutations of these values.\r
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\n" ); document.write( "\n" ); document.write( "Even more amazing is the fact that all this approach can be extended to the systems of four, five and so on unknowns.\r
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\n" ); document.write( "\n" ); document.write( "Thanks to the person who submitted this challenging problem.\r
\n" ); document.write( "\n" ); document.write( "This solution is my gift to you and to the entire community for the day of July, 4.\r
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