document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=995054>Question 995054</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all of the real and imaginary zeros for each polynomial function.\r<BR>\n" );
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document.write( "a. g(x) = x^3 - 2x^2 - 5x + 6\r<BR>\n" );
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document.write( "b. m(x) = x^3 + 4x^2 + 4x + 3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #613903 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=613903\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+x%5E3-2x%5E2+-5x+%2B+6 ALIGN=MIDDLE  ALT=\"g%28x%29+=+x%5E3-2x%5E2+-5x+%2B+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >......set <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=0 ALIGN=MIDDLE  ALT=\"g%28x%29+=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and factor completely\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+x%5E3-2x%5E2%2Bx+-6x+%2B+6 ALIGN=MIDDLE  ALT=\"0+=+x%5E3-2x%5E2%2Bx+-6x+%2B+6\"   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=0+=+%28x%5E3-2x%5E2%2Bx%29+-%286x+-+6%29 ALIGN=MIDDLE  ALT=\"0+=+%28x%5E3-2x%5E2%2Bx%29+-%286x+-+6%29\"   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=0=+x%28x%5E2-2x%2B1%29+-6%28x+-+1%29 ALIGN=MIDDLE  ALT=\"0=+x%28x%5E2-2x%2B1%29+-6%28x+-+1%29\"   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=0=+x%28x-1%29%5E2+-6%28x+-+1%29 ALIGN=MIDDLE  ALT=\"0=+x%28x-1%29%5E2+-6%28x+-+1%29\"   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=0+=+%28x%28x-1%29+-6%29%28x+-+1%29 ALIGN=MIDDLE  ALT=\"0+=+%28x%28x-1%29+-6%29%28x+-+1%29\"   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=0=+%28x%5E2-x+-6%29%28x+-+1%29 ALIGN=MIDDLE  ALT=\"0=+%28x%5E2-x+-6%29%28x+-+1%29\"   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=0+=+%28x%5E2-3x%2B2x+-6%29%28x+-+1%29 ALIGN=MIDDLE  ALT=\"0+=+%28x%5E2-3x%2B2x+-6%29%28x+-+1%29\"   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=0=+%28x%28x-3%29%2B2%28x+-3%29%29%28x+-+1%29 ALIGN=MIDDLE  ALT=\"0=+%28x%28x-3%29%2B2%28x+-3%29%29%28x+-+1%29\"   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=0+=+%28x-3%29+%28x-1%29+%28x%2B2%29 ALIGN=MIDDLE  ALT=\"0+=+%28x-3%29+%28x-1%29+%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "solutions:\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+%28x-3%29+ ALIGN=MIDDLE  ALT=\"0+=+%28x-3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><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;\" ><BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=++%28x-1%29+ ALIGN=MIDDLE  ALT=\"0+=++%28x-1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=++%28x%2B2%29 ALIGN=MIDDLE  ALT=\"0+=++%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, here you have three real zeros\r<BR>\n" );
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document.write( "see it on a graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E3-2x%5E2+-5x+%2B+6%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E3-2x%5E2+-5x+%2B+6%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "b. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%28x%29+=+x%5E3+%2B+4x%5E2+%2B+4x+%2B+3 ALIGN=MIDDLE  ALT=\"m%28x%29+=+x%5E3+%2B+4x%5E2+%2B+4x+%2B+3\"   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=0+=+x%5E3+%2B+3x%5E2%2Bx%5E2+%2B+3x%2Bx+%2B+3 ALIGN=MIDDLE  ALT=\"0+=+x%5E3+%2B+3x%5E2%2Bx%5E2+%2B+3x%2Bx+%2B+3\"   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=0+=+%28x%5E3+%2B+3x%5E2%29%2B%28x%5E2+%2B3x%29%2B+%28x+%2B+3%29 ALIGN=MIDDLE  ALT=\"0+=+%28x%5E3+%2B+3x%5E2%29%2B%28x%5E2+%2B3x%29%2B+%28x+%2B+3%29\"   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=0+=+x%5E2%28x+%2B+3%29%2Bx%28x+%2B3%29%2B+%28x+%2B+3%29 ALIGN=MIDDLE  ALT=\"0+=+x%5E2%28x+%2B+3%29%2Bx%28x+%2B3%29%2B+%28x+%2B+3%29\"   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=0+=+%28x%2B3%29+%28x%5E2%2Bx%2B1%29 ALIGN=MIDDLE  ALT=\"0+=+%28x%2B3%29+%28x%5E2%2Bx%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "solutions:\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+%28x%2B3%29+ ALIGN=MIDDLE  ALT=\"0+=+%28x%2B3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><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;\" >=> one real solution\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+%28x%5E2%2Bx%2B1%29 ALIGN=MIDDLE  ALT=\"0+=+%28x%5E2%2Bx%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >...use quadratic formula\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( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1+%2B-+sqrt%28+1%5E2-4%2A1%2A1+%29%29%2F%282%2A1%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-1+%2B-+sqrt%28+1%5E2-4%2A1%2A1+%29%29%2F%282%2A1%29+\"   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+=+%28-1+%2B-+sqrt%28+1-4+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"x+=+%28-1+%2B-+sqrt%28+1-4+%29%29%2F2+\"   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+=+%28-1+%2B-+sqrt%28+-3+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"x+=+%28-1+%2B-+sqrt%28+-3+%29%29%2F2+\"   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+=+%28-1+%2B-+i%2Asqrt%28+3+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"x+=+%28-1+%2B-+i%2Asqrt%28+3+%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "so, you have two imaginary zeros\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1+%2B+i%2Asqrt%28+3+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"x+=+%28-1+%2B+i%2Asqrt%28+3+%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1+-+i%2Asqrt%28+3+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"x+=+%28-1+-+i%2Asqrt%28+3+%29%29%2F2+\"   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=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E3+%2B+4x%5E2+%2B+4x+%2B+3%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E3+%2B+4x%5E2+%2B+4x+%2B+3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );