document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1011139>Question 1011139</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find an equation for f(x), the polynomial of smallest degree with real coefficients such that f(x) bounces off of the x-axis at &#8722;5, breaks through the x-axis at &#8722;2, has complex roots of &#8722;2+2i and &#8722;4&#8722;2i and passes through the point (0,&#8722;54).  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #626747 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39632)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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=626747\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Real roots are -5 and -2;<BR>\n" );
document.write( "Other complex roots are  -2+2i, -2-2i, -4-2i, -4+2i.<BR>\n" );
document.write( "Further, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=a%2A%28polynomialDegree6%29 ALIGN=MIDDLE  ALT=\"f%28x%29=a%2A%28polynomialDegree6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you use the y-intercept to finally find a.\r<BR>\n" );
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document.write( "Start with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=a%28x%2B5%29%28x%2B2%29%28x-%28-2%2B2i%29%29%28x-%28-2-2i%29%29%28x-%28-4-2i%29%29%28x-%28-4%2B2i%29%29 ALIGN=MIDDLE   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=a%28x%2B5%29%28x%2B2%29%28x%2B2-2i%29%28x%2B2%2B2i%29%28x%2B4%2B2i%29%28x%2B4-2i%29 ALIGN=MIDDLE  ALT=\"a%28x%2B5%29%28x%2B2%29%28x%2B2-2i%29%28x%2B2%2B2i%29%28x%2B4%2B2i%29%28x%2B4-2i%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=a%28x%2B5%29%28x%2B2%29%28%28x%2B2%29%5E2%2B4%29%28%28x%2B4%29%5E2%2B4%29 ALIGN=MIDDLE  ALT=\"a%28x%2B5%29%28x%2B2%29%28%28x%2B2%29%5E2%2B4%29%28%28x%2B4%29%5E2%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >----this really combined two steps including knowing about difference of squares and of i*i=-1;\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x%2B5%29%28x%2B2%29%28x%5E2%2B4x%2B4%2B4%29%28x%5E2%2B8x%2B16%2B4%29 ALIGN=MIDDLE  ALT=\"a%28x%2B5%29%28x%2B2%29%28x%5E2%2B4x%2B4%2B4%29%28x%5E2%2B8x%2B16%2B4%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=a%28x%2B5%29%28x%2B2%29%28x%5E2%2B4x%2B8%29%28x%5E2%2B8x%2B20%29 ALIGN=MIDDLE  ALT=\"a%28x%2B5%29%28x%2B2%29%28x%5E2%2B4x%2B8%29%28x%5E2%2B8x%2B20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--------at this stage, the function is in purely factored form without showing complex nor imaginary expressions - only real coefficients.  You CAN still use this to find the value for a, making use of the given y-intercept; and you may still want to continue the full multiplication into general form. </SPAN>\n" );
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