document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=88390>Question 88390</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello I am currently trying to figure out this problem.  \r<BR>\n" );
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document.write( "Find all real and imaginary zeros for each polynomial function\r<BR>\n" );
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document.write( "f(x)=x^3-27\r<BR>\n" );
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document.write( "This is how I started working out the problem, I have no examples to really work with on this style of problem so I am not to sure if I have even started it correctly.\r<BR>\n" );
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document.write( "f(x)=x^3-27<BR>\n" );
document.write( "x^3-27=0<BR>\n" );
document.write( "   +27=+27<BR>\n" );
document.write( "x^3=27\r<BR>\n" );
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document.write( "I am stuck and not sure where to go next. Your help at solving this problem will be greatly appreciated.\r<BR>\n" );
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document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #64189 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=64189\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-27=0 ALIGN=MIDDLE  ALT=\"x%5E3-27=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%28x%5E2%2B3x%2B9%29=0 ALIGN=MIDDLE  ALT=\"%28x-3%29%28x%5E2%2B3x%2B9%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor using the <a  rel=nofollow HREF=http://www.purplemath.com/modules/specfact2.htm>difference of cubes </a>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3=0 ALIGN=MIDDLE  ALT=\"x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the first factor equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 3 to both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3x%2B9=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B3x%2B9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the second factor equal to zero\r<BR>\n" );
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document.write( "Now let's use the quadratic formula to solve for x:\r<BR>\n" );
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document.write( "Starting with the general quadratic\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the general solution using the quadratic equation is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+%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" );
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document.write( "So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3%2Ax%2B9=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B3%2Ax%2B9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=3 ALIGN=MIDDLE  ALT=\"b=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=9 ALIGN=MIDDLE  ALT=\"c=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-3+%2B-+sqrt%28+%283%29%5E2-4%2A1%2A9+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-3+%2B-+sqrt%28+%283%29%5E2-4%2A1%2A9+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=1, b=3, and c=9\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-3+%2B-+sqrt%28+9-4%2A1%2A9+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-3+%2B-+sqrt%28+9-4%2A1%2A9+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 3 to get 9\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-3+%2B-+sqrt%28+9%2B-36+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-3+%2B-+sqrt%28+9%2B-36+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A9%2A1 ALIGN=MIDDLE  ALT=\"-4%2A9%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-36 ALIGN=MIDDLE  ALT=\"-36\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-3+%2B-+sqrt%28+-27+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-3+%2B-+sqrt%28+-27+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms in the radicand (everything under the square root)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-3+%2B-+3%2Ai%2Asqrt%283%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-3+%2B-+3%2Ai%2Asqrt%283%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root (note: since we cannot take the square root of a negative value, we must factor out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i=sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"i=sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to make the radicand positive)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-3+%2B-+3%2Ai%2Asqrt%283%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%28-3+%2B-+3%2Ai%2Asqrt%283%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 1 to get 2\r<BR>\n" );
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document.write( "After simplifying, the quadratic has roots of\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1.5+%2B+2.59807621135332i ALIGN=MIDDLE  ALT=\"x=-1.5+%2B+2.59807621135332i\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1.5+-+2.59807621135332i ALIGN=MIDDLE  ALT=\"x=-1.5+-+2.59807621135332i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the polynomial has one real zero:\r<BR>\n" );
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document.write( "and 2 imaginary zeros\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1.5+%2B+2.59807621135332i ALIGN=MIDDLE  ALT=\"x=-1.5+%2B+2.59807621135332i\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1.5+-+2.59807621135332i ALIGN=MIDDLE  ALT=\"x=-1.5+-+2.59807621135332i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice if we graph the polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E3-27 ALIGN=MIDDLE  ALT=\"y=x%5E3-27\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+900%2C+900%2C+-35%2C+10%2C+-35%2C+10%2C+x%5E3-27%29+ ALIGN=MIDDLE  ALT=\"+graph%28+900%2C+900%2C+-35%2C+10%2C+-35%2C+10%2C+x%5E3-27%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E3-27 ALIGN=MIDDLE  ALT=\"y=x%5E3-27\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and we can see that the polynomial has one x-intercept (which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >). Since this polynomial has at most 3 complex roots, there must be 2 imaginary roots (because one is real). So our answer is verified. </SPAN>\n" );
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