document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=139381>Question 139381</A>: <I><SPAN STYLE=\"word-break: break-all;\"> y^3 =125   solve the equation for y. seeing if i am doing this right. my answer was y=5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #101647 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=101647\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Your answer is correct -- as far as it goes.  If your assignment is to find just the real number roots, then you are done.  But if you were asked to completely solve the equation, including finding any complex number roots, you are a long way from done.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The Fundamental Theorem of Algebra tells us a couple of important things in this context.  First, any polynomial equation of degree n has exactly n roots.  You have a 3rd degree equation, so there must be 3 roots.  Furthermore, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a root of the polynomial equation, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-a ALIGN=MIDDLE  ALT=\"x-a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of the polynomial.  That is to say that, given a polynomial function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29 ALIGN=MIDDLE  ALT=\"P%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28a%29=0 ALIGN=MIDDLE  ALT=\"P%28a%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then there exists a polynomial function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q%28x%29 ALIGN=MIDDLE  ALT=\"Q%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=%28x-a%29Q%28x%29 ALIGN=MIDDLE  ALT=\"P%28x%29=%28x-a%29Q%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Knowing that 5 is one of the roots of your equation, and we know this because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28y%29=y%5E3-125 ALIGN=MIDDLE  ALT=\"P%28y%29=y%5E3-125\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%285%29=0 ALIGN=MIDDLE  ALT=\"P%285%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, makes the task of finding the other two roots much simpler.  Solving the general cubic is a horror that I wouldn't wish on anyone, even if I particularly disliked them.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can use the fact that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5 ALIGN=MIDDLE  ALT=\"y-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28y%29=y%5E3-125 ALIGN=MIDDLE  ALT=\"P%28y%29=y%5E3-125\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the process of polynomial long division to find the other factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28y%29 ALIGN=MIDDLE  ALT=\"P%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which will be a quadratic polynomial.  But first we need to put <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28y%29 ALIGN=MIDDLE  ALT=\"P%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > into standard form by putting in placeholders for the zero coefficient terms, thus:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28y%29=y%5E3%2B0y%5E2%2B0y-125 ALIGN=MIDDLE  ALT=\"P%28y%29=y%5E3%2B0y%5E2%2B0y-125\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > goes into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E3 ALIGN=MIDDLE  ALT=\"y%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > times.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the first term of the quotient.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5 ALIGN=MIDDLE  ALT=\"y-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E3-5y%5E2 ALIGN=MIDDLE  ALT=\"y%5E3-5y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Subtract this from the first two terms of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28y%29 ALIGN=MIDDLE  ALT=\"P%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y%5E2 ALIGN=MIDDLE  ALT=\"5y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (remember: change the sign and add).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Bring down the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0y ALIGN=MIDDLE  ALT=\"0y\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > goes into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y%5E2 ALIGN=MIDDLE  ALT=\"5y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y ALIGN=MIDDLE  ALT=\"5y\"   DISABLED_style= \"max-width:90%; height:auto;\" > times.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y ALIGN=MIDDLE  ALT=\"5y\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the second term of the quotient.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y ALIGN=MIDDLE  ALT=\"5y\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5 ALIGN=MIDDLE  ALT=\"y-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y%5E2-25y ALIGN=MIDDLE  ALT=\"5y%5E2-25y\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Subtract this from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y%5E2%2B0y ALIGN=MIDDLE  ALT=\"5y%5E2%2B0y\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25y ALIGN=MIDDLE  ALT=\"25y\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Bring down the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-125 ALIGN=MIDDLE  ALT=\"-125\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > goes into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25y ALIGN=MIDDLE  ALT=\"25y\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25 ALIGN=MIDDLE  ALT=\"25\"   DISABLED_style= \"max-width:90%; height:auto;\" > times.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25 ALIGN=MIDDLE  ALT=\"25\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the third term of the quotient.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25 ALIGN=MIDDLE  ALT=\"25\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5 ALIGN=MIDDLE  ALT=\"y-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25y-125 ALIGN=MIDDLE  ALT=\"25y-125\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Subtract this from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25y-125 ALIGN=MIDDLE  ALT=\"25y-125\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > -- meaning no remainder.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The quotient polynomial is then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q%28y%29=y%5E2%2B5y%2B25 ALIGN=MIDDLE  ALT=\"Q%28y%29=y%5E2%2B5y%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the zeros of this polynomial are the other two zeros of your original polynomial.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Using the quadratic formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"y+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-5+%2B-+sqrt%28+%285%29%5E2-4%2A%281%29%2A%2825%29+%29%29%2F%282%2A%281%29%29+ ALIGN=MIDDLE  ALT=\"y+=+%28-5+%2B-+sqrt%28+%285%29%5E2-4%2A%281%29%2A%2825%29+%29%29%2F%282%2A%281%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-5+%2B-+sqrt%28+25-100+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"y+=+%28-5+%2B-+sqrt%28+25-100+%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-5+%2B-+sqrt%28+-75+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"y+=+%28-5+%2B-+sqrt%28+-75+%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-5+%2B+5i%2Asqrt%283%29%29%2F2+ ALIGN=MIDDLE  ALT=\"y+=+%28-5+%2B+5i%2Asqrt%283%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-5+-+5i%2Asqrt%283%29%29%2F2+ ALIGN=MIDDLE  ALT=\"y+=+%28-5+-+5i%2Asqrt%283%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i ALIGN=MIDDLE  ALT=\"i\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the imaginary number defined by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E2=-1 ALIGN=MIDDLE  ALT=\"i%5E2=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In summary, your three roots are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=5 ALIGN=MIDDLE  ALT=\"y%5B1%5D=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D+=+%28-5+%2B+5i%2Asqrt%283%29%29%2F2+ ALIGN=MIDDLE  ALT=\"y%5B2%5D+=+%28-5+%2B+5i%2Asqrt%283%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B3%5D+=+%28-5+-+5i%2Asqrt%283%29%29%2F2+ ALIGN=MIDDLE  ALT=\"y%5B3%5D+=+%28-5+-+5i%2Asqrt%283%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );