document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=901852>Question 901852</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A polynomial of degree 4 with real coefficients has -2i and -5+7i as two of its zeros. What are the two remaining zeros for this polynomial?\r<BR>\n" );
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document.write( "I have no clue where to start. All the problems I have done so far give me the polynomial.Please help </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #546982 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=546982\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> complex roots come in pairs.\r<BR>\n" );
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document.write( "the solution of -2i is one of the roots.<BR>\n" );
document.write( "it's conjugate of +2i is the other root.\r<BR>\n" );
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document.write( "the solution of -5+7i is one of the roots.<BR>\n" );
document.write( "it's conjugate of -5-71 is the other root.\r<BR>\n" );
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document.write( "that's a total of 4 roots which is what you should have since the degree of the equaton is fourth degree.\r<BR>\n" );
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document.write( "remember:\r<BR>\n" );
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document.write( "complex roots always come in pairs.\r<BR>\n" );
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document.write( "a complex root is a root with a real part and an imaginary part.\r<BR>\n" );
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document.write( "the real part of -2i is equal to 0\r<BR>\n" );
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document.write( "that root is really 0 - 2i and 0 + 2i\r<BR>\n" );
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document.write( "any time you see just an imaginary part, you can be sure that the real part is equal to 0 and the complex root is really 0 +/- the imaginary part.\r<BR>\n" );
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document.write( "the real part of 0 is not normally shown.\r<BR>\n" );
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document.write( "if we multiply the roots of this equation, you will be able to see the quadratic equation itself.\r<BR>\n" );
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document.write( "you will not, however, see the graph crossing the x-axis since the graph will only cross the x-axis where the real roots are.\r<BR>\n" );
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document.write( "your equation is found by multiplying the factors together.\r<BR>\n" );
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document.write( "the roots of the equation are shown as:\r<BR>\n" );
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document.write( "-2i<BR>\n" );
document.write( "+2i<BR>\n" );
document.write( "-5-7i<BR>\n" );
document.write( "-5+7i\r<BR>\n" );
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document.write( "the factors of the equation are:\r<BR>\n" );
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document.write( "y = (x - 2i) * (x + 2i) * (x - 5 - 7i) * (x - 5 + 7i)\r<BR>\n" );
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document.write( "i multiplied those factors out and got the following equation:\r<BR>\n" );
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document.write( "y = x^4 - 10x^3 + 78x^2 - 40x + 296\r<BR>\n" );
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document.write( "basically, you multiply the complex roots in pairs.\r<BR>\n" );
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document.write( "the i part will cancel out.\r<BR>\n" );
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document.write( "if i did this right, you should get the following:\r<BR>\n" );
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document.write( "(x-2i) * (x+2i) results in x^2 + 4\r<BR>\n" );
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document.write( "(x-5-7i) * (x-5+7i) results in x^2 - 10x + 74\r<BR>\n" );
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document.write( "you then multiply (x^2 + 4) * (x^2 - 10x + 74) to get x^2 - 10x^3 + 78x^2 - 40x + 296.\r<BR>\n" );
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document.write( "the graph of the equation looks like this:\r<BR>\n" );
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document.write( "since all the roots of the equation are complex, the equation does not cross the x-axis.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/091505.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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