document.write( "Question 1183785: Write an equation each of whose roots are 2 less than 3 times the roots of 3x^3 + 10x^2 + 7x - 10 = 0. \n" ); document.write( "

Algebra.Com's Answer #814252 by ikleyn(52838) $\"\"$ $\"About$

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\n" ); document.write( "Write an equation each of whose roots are 2 less than 3 times the roots of 3x^3 + 10x^2 + 7x - 10 = 0
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\n" ); document.write( "\n" ); document.write( " This problem's solution is to apply the Vieta's theorem several times.\r
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document.write( "Let  a,  b  and  c  be the roots of the given equation;\r\n" );
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document.write( "let  u,  v  and  w  be the roots of the projected equation.\r\n" );
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document.write( "According to the condition, we have \r\n" );
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document.write( "    u = 3a-2,  v = 3b-2,  w = 3c-2.\r\n" );
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document.write( "    +----------------------------------------------------------------------+\r\n" );
document.write( "    |   Let the projected equation be  px^3 + qx^2 + rx + s = 0.           |\r\n" );
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document.write( "    |   Our goal is to determine the coefficients p, q, r and s.           |\r\n" );
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document.write( "According to Vieta's theorem,  a + b + c = .\r\n" );
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document.write( "Hence,  u + v + w = (3a-2) + (3b-2) + (3c-2) = 3*(a+b+c) - 6 =  = -10 - 6 = -16.\r\n" );
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document.write( "Thus     = 16,  according to Vieta's theorem.\r\n" );
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document.write( "Next, according to Vieta's theorem,  ab + ac + bc = .\r\n" );
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document.write( "Hence,  uv + uw + vw = (3a-2)*(3b-2) + (3a-2)*(3c-2) + (3b-2)*(3c-2) = 9(ab + ac + bc) - (6a + 6b + 6a + 6c + 6b + 6c) + (4+4+4) = \r\n" );
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document.write( "                     = 9(ab + ac + bc) - 12(a+b+c) + 12 =  = 21 + 40 + 12 = 73.\r\n" );
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document.write( "Thus     = 73,  according to Vieta's theorem.\r\n" );
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document.write( "Finally, according to Vieta's theorem,  abc = {{10/3}}}.\r\n" );
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document.write( "Hence,  uvw = (3a-2)*(3b-2)*(3c-2) = 27abc - 18(ab + ac + bc) + 12(a + b + c) - 8 =  \r\n" );
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document.write( "                     =  = 9*10 - 6*7 - 4*10 - 8 = 0.\r\n" );
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document.write( "Thus     = 0,  according to Vieta's theorem.\r\n" );
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document.write( "Since the projected polynomial coefficients ratios are integer numbers, we can take p = 1.\r\n" );
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document.write( "It gives q = 16,  r = 73,  s = 0.\r\n" );
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document.write( "Thus the projected equation  is  x^3 + 16x^2 + 73x = 0.      ANSWER\r\n" );
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The post-solution note

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\n" ); document.write( "\n" ); document.write( "            The solution by @MathLover1 works due to that happy occasion, that the given equation has a rational root.\r
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\n" ); document.write( "\n" ); document.write( "            My approach / (my solution) works independently of this occasion.\r
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\n" ); document.write( "\n" ); document.write( "            With my approach, there is no need in finding the roots of the given polynomial.\r
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