document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199426>Question 1199426</A>: <I><SPAN STYLE=\"word-break: break-all;\"> let the roots of the equation x^3 -2x^2 -3x-7=0 be r, s, and t. find the equation whose roots are  r^2, s^2 and t^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833373 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=833373\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "r, s, and t are roots of a cubic.<BR>\n" );
document.write( "Which means x-r, x-s, x-t are factors.\r<BR>\n" );
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document.write( "Multiply those factors and expand like so.<BR>\n" );
document.write( "(x-r)(x-s)(x-t)<BR>\n" );
document.write( "(x-r)(x^2-tx-sx+st)<BR>\n" );
document.write( "x(x^2-tx-sx+st)-r(x^2-tx-sx+st)<BR>\n" );
document.write( "(x^3-tx^2-sx^2+stx)+(-rx^2+rtx+rsx-rst)<BR>\n" );
document.write( "x^3-tx^2-sx^2+stx-rx^2+rtx+rsx-rst<BR>\n" );
document.write( "x^3+(-tx^2-sx^2-rx^2)+(stx+rtx+rsx)-rst<BR>\n" );
document.write( "x^3-(t+s+r)x^2+(st+rt+rs)x-rst<BR>\n" );
document.write( "x^3-(r+s+t)x^2+(rs+st+rt)x-rst\r<BR>\n" );
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document.write( "In short<BR>\n" );
document.write( "(x-r)(x-s)(x-t) = x^3-(r+s+t)x^2+(rs+st+rt)x-rst<BR>\n" );
document.write( "This is mentioned in Vieta's formulas regarding cubics.\r<BR>\n" );
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document.write( "Compare<BR>\n" );
document.write( "x^3-(r+s+t)x^2+(rs+st+rt)x-rst<BR>\n" );
document.write( "with<BR>\n" );
document.write( "x^3-2x^2-3x-7<BR>\n" );
document.write( "to see that<ul><li>r+s+t = 2</li><li>rs+st+rt = -3</li><li>rst = 7</li></ul>Through similar calculations, we'll have:<BR>\n" );
document.write( "(x-r^2)(x-s^2)(x-t^2) = x^3-(r^2+s^2+t^2)x^2+(r^2s^2+s^2t^2+r^2t^2)x-(rst)^2\r<BR>\n" );
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document.write( "We need to know the following three items:<ul><li>r^2+s^2+t^2</li><li>r^2s^2+s^2t^2+r^2t^2</li><li>rst</li></ul>to be able to determine the cubic equation that has roots r^2, s^2, and t^2.\r<BR>\n" );
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document.write( "One useful polynomial identity is<BR>\n" );
document.write( "(r+s+t)^2 = r^2+s^2+t^2+2(rs+st+rt)<BR>\n" );
document.write( "The proof of which I leave to the reader.\r<BR>\n" );
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document.write( "That identity rearranges to<BR>\n" );
document.write( "r^2+s^2+t^2 = (r+s+t)^2 - 2(rs+st+rt)\r<BR>\n" );
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document.write( "Then we plug in the items mentioned earlier<BR>\n" );
document.write( "r^2+s^2+t^2 = (r+s+t)^2 - 2(rs+st+rt)<BR>\n" );
document.write( "r^2+s^2+t^2 = (2)^2 - 2(-3)<BR>\n" );
document.write( "r^2+s^2+t^2 = 10\r<BR>\n" );
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document.write( "With similar steps we'll have<BR>\n" );
document.write( "(rs+st+rt)^2 = r^2s^2+s^2t^2+r^2t^2+2(rs^2t+rst^2+r^2st)<BR>\n" );
document.write( "r^2s^2+s^2t^2+r^2t^2 = (rs+st+rt)^2 - 2(rs^2t+rst^2+r^2st)<BR>\n" );
document.write( "r^2s^2+s^2t^2+r^2t^2 = (rs+st+rt)^2 - 2rst(r+s+t)<BR>\n" );
document.write( "r^2s^2+s^2t^2+r^2t^2 = (-3)^2 - 2*7*(2)<BR>\n" );
document.write( "r^2s^2+s^2t^2+r^2t^2 = -19\r<BR>\n" );
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document.write( "We found the following<ul><li>r^2+s^2+t^2 = 10</li><li>r^2s^2+s^2t^2+r^2t^2 = -19</li><li>rst = 7</li></ul>and now have enough information.<BR>\n" );
document.write( "Let's plug in those items to get the following.<BR>\n" );
document.write( "x^3-(r^2+s^2+t^2)x^2+(r^2s^2+s^2t^2+r^2t^2)x-(rst)^2<BR>\n" );
document.write( "x^3-10x^2+(-19)x-(7)^2<BR>\n" );
document.write( "<font color=red>x^3-10x^2-19x-49</font>\r<BR>\n" );
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document.write( "Answer:<BR>\n" );
document.write( "If x^3-2x^2-3x-7=0 has roots {r,s,t}, then the equation that has roots {r^2,s^2,t^2} is <font color=red size=4>x^3-10x^2-19x-49=0</font><BR>\n" );
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