document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209729>Question 1209729</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let t be a root of f(x) = x^3 - x + 3.  Evaluate t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850045 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52778)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850045\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Let t be a root of f(x) = x^3 - x + 3.  Evaluate t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13.<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The answer in the post by @CPhill is incorrect.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;After detecting his erroneous answer, I did a thorough analysis and check.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;My results are shown below.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I disproved incorrect answer to @CPhill and produced a correct solution/answer.\r<BR>\n" );
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document.write( "From the condition, we have t^3 - t + 3 = 0,  or  t^3 = t-3.\r\n" );
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document.write( "It allows to reduce the degrees in the given expression  t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13.\r\n" );
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document.write( "We have \r\n" );
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document.write( "    t^6 = t^3*t^3 = (t-3)*(t-3) = t^2 - 6t + 9,\r\n" );
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document.write( "    t^5 = t^2*t^3 = t^2*(t-3) = t^3 - 3t^2 = (t-3)- 3t^2 = -3t^2 + t - 3,\r\n" );
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document.write( "    t^4 = t*t^3 = t*(t-3) = t^2 - 3t.\r\n" );
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document.write( "Now we substitute these reduced expressions for t^6, t^5 and t^4 into the given expression. We get\r\n" );
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document.write( "    t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13 = (t^2 - 6t + 9) - 4*(-3t^2 + t - 3) + 7*(t^2 - 3t) - 3t^2 + 10t - 13 = \r\n" );
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document.write( "    = now combine like terms = (t^2 + 12t^2 + 7t^2 - 3t^2) + (-6t - 4t - 21t + 10t) + (9 + 12 - 13) = \r\n" );
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document.write( "    = 17t^2 - 21t + 8.\r\n" );
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document.write( "At this point, we can not simplify further.\r\n" );
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document.write( "    |   At this point, after obtaining the same reduced expression,            |\r\n" );
document.write( "    |   @CPhill in his post produced WRONG answer 8 for the expression value.  |\r\n" );
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document.write( "So, to find the value of t, we should solve this equation x^3 - x + 3 = 0.\r\n" );
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document.write( "It has no rational roots.  So, I used the online solver \r\n" );
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document.write( "https://www.mathportal.org/calculators/solving-equations/polynomial-equation-solver.php\r\n" );
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document.write( "to find a/the real root.\r\n" );
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document.write( "The solver determined that this polynomial has one real root (which is not rational) and two complex roots.\r\n" );
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document.write( "                The real root is  t = -1.6717 (approximately)\r\n" );
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document.write( "See the solution by the online solver at this link\r\n" );
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document.write( "https://www.mathportal.org/calculators/solving-equations/polynomial-equation-solver.php?val1=x%5E3+-+x+%2B+3&val2=0&val3=%7Bx%7D%5E%7B3%7D-x%2B3%3D0 )\r\n" );
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document.write( "        Let's check this root  (-1.6717)^3 - (-1.6717) + 3 = -8.73813E-07,\r\n" );
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document.write( "              so we can accept that it is the right real root.\r\n" );
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document.write( "Then the value of our final reduced expression is  17*(-1.6717)^2 - 21*(-1.6717) + 8 = 90.61357513.\r\n" );
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document.write( "It is just almost the answer to the problem.\r\n" );
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document.write( "To make check, I substitute t = -1.6717 into the given expression and calculate \r\n" );
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document.write( "    (-1.6717)^6 - 4*(-1.6717)^5 + 7*(-1.6717)^4 - 3*(-1.6717)^2 + 10*(-1.6717) - 13 = 90.61360678.\r\n" );
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document.write( "With good/reasonable precision, we can accept this check.\r\n" );
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document.write( "So, the correct answer (=the correct evaluation for the expression) is 90.6136 (approximately)\r<BR>\n" );
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document.write( "The answer by @CPhill, giving the value of 8, is incorrect.\r<BR>\n" );
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document.write( "Couple of words after completing the solution // post-solution notes.\r<BR>\n" );
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document.write( "In this problem, using the method of reducing degree does not lead to a nice solution.\r<BR>\n" );
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document.write( "The problem can be solved, with the same success, by the method of brute force<BR>\n" );
document.write( "(solving equation approximately using calculator or online solvers and substituting the root into the expression).\r<BR>\n" );
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document.write( "In that sense, &nbsp;the problem contains &nbsp;NOTHING &nbsp;except making stupid calculations,<BR>\n" );
document.write( "and, &nbsp;THEREFORE, &nbsp;has &nbsp;ZERO &nbsp;educational value.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Regarding the post by @CPhill . . . \r<BR>\n" );
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document.write( "Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.\r<BR>\n" );
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document.write( "The artificial intelligence is like a baby now. It is in the experimental stage <BR>\n" );
document.write( "of development and can make mistakes and produce nonsense without any embarrassment.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It has no feeling of shame - it is shameless.\r<BR>\n" );
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document.write( "This time, again, &nbsp;it made an error.\r<BR>\n" );
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document.write( "Although the @CPhill' solution are copy-paste &nbsp;Google &nbsp;AI solutions, &nbsp;there is one essential difference.\r<BR>\n" );
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document.write( "Every time, &nbsp;Google &nbsp;AI &nbsp;makes a note at the end of its solutions that &nbsp;Google &nbsp;AI &nbsp;is experimental<BR>\n" );
document.write( "and can make errors/mistakes.  &nbsp;&nbsp;So, &nbsp;in this sense, &nbsp;Google &nbsp;AI &nbsp;is honest.\r<BR>\n" );
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document.write( "All @CPhill' solutions are copy-paste of &nbsp;Google &nbsp;AI &nbsp;solutions, with one difference:<BR>\n" );
document.write( "@PChill never makes this notice and never says that his solutions are copy-past that of Google.<BR>\n" );
document.write( "So, &nbsp;in this sense, &nbsp;@CPhill &nbsp;makes a dishonest business at this forum.\r<BR>\n" );
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document.write( "Every time, &nbsp;@CPhill embarrassed to tell the truth.\r<BR>\n" );
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document.write( "But I am not embarrassing to tell the truth, &nbsp;as it is my duty at this forum.\r<BR>\n" );
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document.write( "And the last my comment.\r<BR>\n" );
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document.write( "When you obtain such posts from @CPhill, &nbsp;remember, &nbsp;that &nbsp;NOBODY &nbsp;is responsible for their correctness, <BR>\n" );
document.write( "until the specialists and experts will check and confirm their correctness.\r<BR>\n" );
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document.write( "Without it, &nbsp;their reliability is &nbsp;ZERO and their creadability is &nbsp;ZERO, &nbsp;too.\r<BR>\n" );
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