document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208646>Question 1208646</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Form a polynomial f left parenthesis x right parenthesis with real coefficients having the given degree and zeros.<BR>\n" );
document.write( "Degree​ 5;  ​ zeros: negative 4 ; minus i ; 5 plus i </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847031 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=847031\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer is  <font color=red>f(x) = x^5 - 6x^4 - 13x^3 + 98x^2 - 14x + 104</font>\r<BR>\n" );
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document.write( "How to find that answer:\r<BR>\n" );
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document.write( "Since all of the coefficients are real numbers, this means the complex roots come in conjugate pairs.<BR>\n" );
document.write( "a+bi pairs with a-bi\r<BR>\n" );
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document.write( "-i pairs with i<BR>\n" );
document.write( "5+i pairs with 5-i\r<BR>\n" );
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document.write( "The five roots are: -4, i, -i, 5+i, 5-i<BR>\n" );
document.write( "Recall the fundamental theorem of algebra says that any n<sup>th</sup> degree polynomial has n complex roots.\r<BR>\n" );
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document.write( "x = i has both sides square to x^2 = -1 and then we get everything to one side: x^2+1=0. So that yields the factor (x^2+1).\r<BR>\n" );
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document.write( "x = 5+i becomes x-5 = i; then both sides square to x^2-10x+25 = -1 and it becomes x^2-10x+26=0<BR>\n" );
document.write( "Use the quadratic formula to solve x^2-10x+26=0 and you should get x = 5+i and x = 5-i.<BR>\n" );
document.write( "Note: If you're using GeoGebra, you need to use the <a  rel=nofollow HREF=\"https://geogebra.github.io/docs/manual/en/commands/CSolve/\">CSolve</a> command (in contrast to the regular <a  rel=nofollow HREF=\"https://geogebra.github.io/docs/manual/en/commands/Solve/\">Solve</a> command). Otherwise, it will produce an empty set of solutions.\r<BR>\n" );
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document.write( "That slight tangent aside, we can say the following<BR>\n" );
document.write( "The root x = -4 leads to the factor (x+4)<BR>\n" );
document.write( "The roots x = i, x = -i lead to the factor (x^2+1)<BR>\n" );
document.write( "The roots x = 5+i, x = 5-i lead to the factor (x^2-10x+26)\r<BR>\n" );
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document.write( "The goal is to expand this out:  (x+4)(x^2+1)(x^2-10x+26)\r<BR>\n" );
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document.write( "--------------------------------------------------------------------------\r<BR>\n" );
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document.write( "For now let's focus on expanding the portion (x+4)(x^2+1)\r<BR>\n" );
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document.write( "We could use the FOIL rule, but I'll use the <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/playground/lessons/box-method.lesson\">box method</a> instead.<BR>\n" );
document.write( "Place the terms x and 4 along the left hand side. Place the terms x^2 and 1 along the top.<BR>\n" );
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document.write( "<tr><TD ></TD><TD >x^2</TD><TD >1</TD></tr><tr><TD >x</TD><TD ></TD><TD ></TD></tr><tr><TD >4</TD><TD ></TD><TD ></TD></tr>\n" );
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document.write( "To fill out this table, multiply each pair of headers. <BR>\n" );
document.write( "Eg: top left corner is x^2*x = <font color=blue>x^3</font><BR>\n" );
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document.write( "<tr><TD ></TD><TD >x^2</TD><TD >1</TD></tr><tr><TD >x</TD><TD ><font color=blue>x^3</font></TD><TD ><font color=blue>x</font></TD></tr><tr><TD >4</TD><TD ><font color=blue>4x^2</font></TD><TD ><font color=blue>4</font></TD></tr>\n" );
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document.write( "The inner terms in <font color=blue>blue</font> are then added to get x^3+4x^2+x+4. There aren't any like terms to combine.\r<BR>\n" );
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document.write( "We have shown that (x+4)(x^2+1) = x^3+4x^2+x+4\r<BR>\n" );
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document.write( "So <BR>\n" );
document.write( "(x+4)(x^2+1)(x^2-10x+26)<BR>\n" );
document.write( "updates to<BR>\n" );
document.write( "(x^3+4x^2+x+4)(x^2-10x+26)\r<BR>\n" );
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document.write( "We'll need to do one more application of the box method.<BR>\n" );
document.write( "Here's the blank template with the headers filled in.<BR>\n" );
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document.write( "<tr><TD ></TD><TD >x^2</TD><TD >-10x</TD><TD >26</TD></tr><tr><TD >x^3</TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >4x^2</TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >x</TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >4</TD><TD ></TD><TD ></TD><TD ></TD></tr>\n" );
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document.write( "And here's the completed table<BR>\n" );
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document.write( "<tr><TD ></TD><TD >x^2</TD><TD >-10x</TD><TD >26</TD></tr><tr><TD >x^3</TD><TD ><font color=blue>x^5</font></TD><TD ><font color=blue>-10x^4</font></TD><TD ><font color=blue>26x^3</font></TD></tr><tr><TD >4x^2</TD><TD ><font color=blue>4x^4</font></TD><TD ><font color=blue>-40x^3</font></TD><TD ><font color=blue>104x^2</font></TD></tr><tr><TD >x</TD><TD ><font color=blue>x^3</font></TD><TD ><font color=blue>-10x^2</font></TD><TD ><font color=blue>26x</font></TD></tr><tr><TD >4</TD><TD ><font color=blue>4x^2</font></TD><TD ><font color=blue>-40x</font></TD><TD ><font color=blue>104</font></TD></tr>\n" );
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document.write( "Add up the terms in blue. <BR>\n" );
document.write( "This time we have groups of like terms to combine (eg: 4x^4 + (-10x^4) = -6x^4)<BR>\n" );
document.write( "Notice the like terms are along northeast diagonals. <BR>\n" );
document.write( "I'll skip a bit of scratch work and leave it to the student. \r<BR>\n" );
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document.write( "You should get the final result <font color=red>f(x) = x^5 - 6x^4 - 13x^3 + 98x^2 - 14x + 104</font>\r<BR>\n" );
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document.write( "You can use various software tools to verify this answer.<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.wolframalpha.com/input?i=x%5E5+-+6x%5E4+-+13x%5E3+%2B+98x%5E2+-+14x+%2B+104%3D0\">WolframAlpha</a> is one example. <a  rel=nofollow HREF=\"https://www.geogebra.org/\">GeoGebra</a> is another (make sure to use Csolve instead of Solve).<BR>\n" );
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