document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183785>Question 1183785</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write an equation each of whose roots are 2 less than 3 times the roots of 3x^3 + 10x^2 + 7x - 10 = 0.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814256 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=814256\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> *\r<BR>\n" );
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document.write( "The two prior tutors took great details to solve the given problem.  \r<BR>\n" );
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document.write( "But in fact, the solution is just a simple exercise in transformation of variables.\r<BR>\n" );
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document.write( "Suppose that the roots \"r\" of the equation is transformed into some form g(r).  <BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha%2A%28X+-+g%28r%5B1%5D%29%29%28X+-+g%28r%5B2%5D%29%29%2A%22...%22%2A%28X-g%28r%5Bn%5D%29%29+=+0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > be the new equation after the roots are transformed.  <BR>\n" );
document.write( "Then we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X+=+g%28x%29 ALIGN=MIDDLE  ALT=\"X+=+g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+g%5E%28-1%29%28X%29 ALIGN=MIDDLE  ALT=\"x+=+g%5E%28-1%29%28X%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, as long as the inverse function exists and is well-defined.  <BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X+-+g%28r%5Bi%5D%29+=+0 ALIGN=MIDDLE  ALT=\"X+-+g%28r%5Bi%5D%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > when the roots are being obtained, we get \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X+=+g%28r%5Bi%5D%29 ALIGN=MIDDLE  ALT=\"X+=+g%28r%5Bi%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%5E%28-1%29%28X%29+=+r%5Bi%5D ALIGN=MIDDLE  ALT=\"g%5E%28-1%29%28X%29+=+r%5Bi%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%5E%28-1%29%28X%29+-+r%5Bi%5D+=+x-r%5Bi%5D=0 ALIGN=MIDDLE  ALT=\"g%5E%28-1%29%28X%29+-+r%5Bi%5D+=+x-r%5Bi%5D=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which brings back the roots and the linear factors of the original equation. <BR>\n" );
document.write( "Hence, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta%2A%28g%5E%28-1%29%28X%29+-+r%5B1%5D%29%2A%28g%5E%28-1%29%28X%29+-+r%5B2%5D%29%2A%22...%22%2A%28g%5E%28-1%29%28X%29+-+r%5Bn%5D%29+%0D%0A=+c%5Bn%5D%2AX%5En+%2B+c%5Bn-1%5D%2AX%5E%28n-1%29+%2B%22...%22%2B+c%5B1%5D%2AX+%2B+c%5B0%5D=+0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "becomes the required equation under the new transformation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X+=+g%28x%29 ALIGN=MIDDLE  ALT=\"X+=+g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "In other words, we only have to find the inverse function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X+=+g%28x%29+=+3x-2 ALIGN=MIDDLE  ALT=\"X+=+g%28x%29+=+3x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%5E%28-1%29%28X%29+=+%28X%2B2%29%2F3 ALIGN=MIDDLE  ALT=\"g%5E%28-1%29%28X%29+=+%28X%2B2%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and substitute into the original equation.\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A%28%28X%2B2%29%2F3%29%5E3+%2B+10%2A%28%28X%2B2%29%2F3+%29%5E2+%2B+7%2A%28%28X%2B2%29%2F3+%29+-+10+=+X%5E3+%2B+16X%5E2+%2B+73X+=+0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > is the new transformed equation.  <BR>\n" );
document.write( "Then just switch back to the usual \"x\" variable, giving <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+%2B+16x%5E2+%2B+73x+=+0 ALIGN=MIDDLE  ALT=\"x%5E3+%2B+16x%5E2+%2B+73x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Note that this result is true whenever the transformation g(x) has a well-defined inverse.  \r<BR>\n" );
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document.write( "Also, under a general g(x), the resulting equation may not be a polynomial, as what happens when one wants to find the equation <BR>\n" );
document.write( "whose roots are cubes of the roots of the original equation. The equation will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x+%2B+10x%5E%282%2F3%29+%2B+7x%5E%281%2F3%29+-+10+=+0 ALIGN=MIDDLE  ALT=\"3x+%2B+10x%5E%282%2F3%29+%2B+7x%5E%281%2F3%29+-+10+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "So no need for hanky panky solutions that don't address the heart of the matter.  <BR>\n" );
document.write( "Especially from people whose first language is NOT English.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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