document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=73367>Question 73367</A>: <I><SPAN STYLE=\"word-break: break-all;\"> FIND A THIRD-DEGREE POLYNOMIAL EQUATION WITH RATIONAL COEFFICIENTS THAT HAS ROOTS -5 AND 6+i </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #52540 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=52540\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Complex roots occur only in pairs, so the roots are actually 6 - i and 6 + 1 and -5. <BR>\n" );
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document.write( "Putting these into factor form you get:<BR>\n" );
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document.write( "(x + 5)*(x - (6 + i))*(x - (6 - i))<BR>\n" );
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document.write( "All you have to do now is multiply all of these together and you will have the answer.<BR>\n" );
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document.write( "Let's first work on multiplying the two complex terms. First remove the parentheses <BR>\n" );
document.write( "by changing the signs of the terms within.  When you do the two factors that you will be<BR>\n" );
document.write( "multiplying are:<BR>\n" );
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document.write( "(x - 6 - i)* (x - 6 + i)<BR>\n" );
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document.write( "The way to do this is to take the terms in the first set of parentheses one at a time and <BR>\n" );
document.write( "multiply them by the terms in the second set of parentheses.  <BR>\n" );
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document.write( "So select the x in the first set of parentheses and multiply it by the x, then the –6, and then<BR>\n" );
document.write( "the +i from the second set of parentheses to get three answer terms consisting of<BR>\n" );
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document.write( "x^2–6x+xi<BR>\n" );
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document.write( "Then select the –6 in the first set of parentheses and multiply it by the x, then the –6, and<BR>\n" );
document.write( "then the +i from the second set of parentheses to get three more answer terms consisting of <BR>\n" );
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document.write( "-6x + 36 –6i<BR>\n" );
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document.write( "Finally, take the –i from the first set of parentheses and multiply it times the x, the –6, <BR>\n" );
document.write( "and the i in the second set of parentheses to get three more answer terms consisting of<BR>\n" );
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document.write( "-xi + 6i –i^2  <BR>\n" );
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document.write( "One more thing. Recall that by definition i^2 = -1 and<BR>\n" );
document.write( "if you <BR>\n" );
document.write( "substitute –1 for i^2 in the three answer terms of this group, you get<BR>\n" );
document.write( "-xi + 6i –(-1) <BR>\n" );
document.write( "which simplifies to -xi + 6i +1<BR>\n" );
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document.write( "Now add the three groups of answer terms, noting that some of them are equal but with<BR>\n" );
document.write( "opposite signs so they cancel out.  The string of answer terms before cancellation is:<BR>\n" );
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document.write( "x^2 – 6x + xi –6x + 36 + 6i – xi + 6i + 1<BR>\n" );
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document.write( "Cancel the 6i terms and the xi terms and you are left with:<BR>\n" );
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document.write( "x^2 – 6x – 6x  + 36 + 1<BR>\n" );
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document.write( "When you combine like terms you have:<BR>\n" );
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document.write( "x^2 – 12x + 37<BR>\n" );
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document.write( "That takes care of multiplying two of the factors together.  Now all you have to do is<BR>\n" );
document.write( "multiply that trinomial by the third factor of the binomial (x + 5)<BR>\n" );
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document.write( "Multiply each term in the trinomial using the x from the binomial to get:<BR>\n" );
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document.write( " x^3 – 12x^2 + 37x<BR>\n" );
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document.write( "Then multiply each of the terms in the trinomial by the +5 from the binomial to get:<BR>\n" );
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document.write( "5x^2 – 60x  + 185<BR>\n" );
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document.write( "Add these 6 multiplied terms and you get:<BR>\n" );
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document.write( "x^3 - 12x^2 + 37x + 5x^2 – 60x + 185<BR>\n" );
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document.write( "Combine the terms having like powers of x and you end up with:<BR>\n" );
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document.write( "x^3 - 7x^2 – 23x + 185<BR>\n" );
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document.write( "This is the third degree polynomial you were looking for. In equation form it would be:<BR>\n" );
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document.write( "x^3 - 7x^2 – 23x + 185 = 0<BR>\n" );
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document.write( "Hope this is helpful to you and that you can follow your way down this long and winding path. </SPAN>\n" );
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