document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=118101>Question 118101</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The degree three polynomial f(x) with real coefficients and leading coefficient 1, has -3 and +4i among its roots.  Express f(x) as a product of linear and quadratic polynomials with real coefficients. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #86191 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=86191\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Remember, if a polynomial has roots of a and b, then it's factorization will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-a%29%28x-b%29 ALIGN=MIDDLE  ALT=\"%28x-a%29%28x-b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So if a polynomial has roots of -3, 4i, and -4i (remember complex roots always come in pairs) , then it's factorization will be \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-%28-3%29%29%28x-4i%29%28x-%28-4i%29%29 ALIGN=MIDDLE  ALT=\"%28x-%28-3%29%29%28x-4i%29%28x-%28-4i%29%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=%28x%2B3%29%28x-4i%29%28x%2B4i%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%28x-4i%29%28x%2B4i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-%28-3%29 ALIGN=MIDDLE  ALT=\"x-%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3 ALIGN=MIDDLE  ALT=\"x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-%28-4i%29 ALIGN=MIDDLE  ALT=\"x-%28-4i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B4i ALIGN=MIDDLE  ALT=\"x%2B4i\"   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%2B3%29%28x%5E2%2B4xi-4xi-4i%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%28x%5E2%2B4xi-4xi-4i%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Foil <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4i%29%28x%2B4i%29 ALIGN=MIDDLE  ALT=\"%28x-4i%29%28x%2B4i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B4xi-4xi-4i%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B4xi-4xi-4i%5E2\"   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%2B3%29%28x%5E2-16i%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%28x%5E2-16i%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%28x%5E2-16%28-1%29%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%28x%5E2-16%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E2 ALIGN=MIDDLE  ALT=\"i%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > with -1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%28x%5E2%2B16%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%28x%5E2%2B16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B3x%5E2%2B16x%2B48 ALIGN=MIDDLE  ALT=\"x%5E3%2B3x%5E2%2B16x%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now foil <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%28x%5E2%2B16%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%28x%5E2%2B16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B3x%5E2%2B16x%2B48 ALIGN=MIDDLE  ALT=\"x%5E3%2B3x%5E2%2B16x%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the polynomial with roots of -3 and +4i is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B3x%5E2%2B16x%2B48 ALIGN=MIDDLE  ALT=\"x%5E3%2B3x%5E2%2B16x%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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