document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201674>Question 1201674</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the given zero to find the remaining zeros of the polynomial function \r<BR>\n" );
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document.write( "P(x) =x^4-6x^3+19x^2-6x+18;  3-3i </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836160 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=836160\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The term \"root\" is another way of saying \"zero of a function\"<BR>\n" );
document.write( "The root is visually represented by the x intercept, assuming the root is a real number.\r<BR>\n" );
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document.write( "In the case of 3-3i being a root, we unfortunately cannot represent this as an x intercept.<BR>\n" );
document.write( "It has a conjugate pair of 3+3i\r<BR>\n" );
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document.write( "We'll take the root x = 3-3i and work backwards, so to speak, to arrive at a quadratic polynomial.\r<BR>\n" );
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document.write( "x = 3-3i<BR>\n" );
document.write( "x-3 = -3i<BR>\n" );
document.write( "(x-3)^2 = (-3i)^2<BR>\n" );
document.write( "(x-3)^2 = 9i^2<BR>\n" );
document.write( "(x-3)^2 = 9(-1)<BR>\n" );
document.write( "(x-3)^2 = -9<BR>\n" );
document.write( "(x-3)^2+9 = 0<BR>\n" );
document.write( "x^2-6x+9+9 = 0<BR>\n" );
document.write( "x^2-6x+18 = 0<BR>\n" );
document.write( "Similar, if not almost identical, steps will apply for the root x = 3+3i.\r<BR>\n" );
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document.write( "If you were to use the quadratic formula to solve x^2-6x+18 = 0, then you should get the complex roots x = 3+3i and x = 3-3i.<BR>\n" );
document.write( "You can use WolframAlpha or the CAS feature in GeoGebra to check your work.\r<BR>\n" );
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document.write( "Because 3-3i is a root of P(x)=x^4-6x^3+19x^2-6x+18, we know that x^2-6x+18 is a factor of P(x).\r<BR>\n" );
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document.write( "P(x) = (x^2-6x+18)*Q(x)<BR>\n" );
document.write( "where Q is some quotient polynomial.<BR>\n" );
document.write( "The remainder is 0 because x^2-6x+18 is a factor.\r<BR>\n" );
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document.write( "Isolating Q means we divide both sides by (x^2-6x+18)<BR>\n" );
document.write( "Q(x) = P(x)/(x^2-6x+18)<BR>\n" );
document.write( "Q(x) = (x^4-6x^3+19x^2-6x+18)/(x^2-6x+18)\r<BR>\n" );
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document.write( "From here we use polynomial long division to determine Q.<BR>\n" );
document.write( "Synthetic division will <u>not</u> work because the denominator x^2-6x+18 is not linear.\r<BR>\n" );
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document.write( "Here is what the polynomial long division will look like.<BR>\n" );
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document.write( "We therefore find that<BR>\n" );
document.write( "Q(x) = x^2+1<BR>\n" );
document.write( "P(x) = Q(x)*(x^2-6x+18)<BR>\n" );
document.write( "P(x) = (x^2+1)(x^2-6x+18)\r<BR>\n" );
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document.write( "In other words,<BR>\n" );
document.write( "(x^2+1)(x^2-6x+18) = x^4-6x^3+19x^2-6x+18<BR>\n" );
document.write( "is an identity. The equation is true for all values of x.\r<BR>\n" );
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document.write( "We can confirm this by use of the <a  rel=nofollow HREF= \"https://www.algebra.com/algebra/homework/playground/lessons/box-method.lesson\">box method</a><BR>\n" );
document.write( "We'll have the terms of x^2+1 along the left side and the terms of x^2-6x+18 along the top<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >x^2</TD><TD >-6x</TD><TD >18</TD></tr><tr><TD >x^2</TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >1</TD><TD ></TD><TD ></TD><TD ></TD></tr>\n" );
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document.write( "Then to fill out each inner cell, multiply the left and top headers.<BR>\n" );
document.write( "Example: x^2 times x^2 = x^4 in the upper left corner<BR>\n" );
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document.write( "<tr><TD ></TD><TD >x^2</TD><TD >-6x</TD><TD >18</TD></tr><tr><TD >x^2</TD><TD >x^4</TD><TD >-6x^3</TD><TD >18x^2</TD></tr><tr><TD >1</TD><TD >x^2</TD><TD >-6x</TD><TD >18</TD></tr>\n" );
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document.write( "Add up the terms inside those cells we filled out<BR>\n" );
document.write( "x^4-6x^3+18x^2+x^2-6x+18 = x^4-6x^3+19x^2-6x+18<BR>\n" );
document.write( "We have confirmed that (x^2+1)(x^2-6x+18) expands to x^4-6x^3+19x^2-6x+18<BR>\n" );
document.write( "In other words, we have confirmed x^4-6x^3+19x^2-6x+18 factors to (x^2+1)(x^2-6x+18)\r<BR>\n" );
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document.write( "As stated earlier, the two roots x = 3+3i and x = 3-3i were from solving x^2-6x+18=0.\r<BR>\n" );
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document.write( "To determine the other two missing other roots, we solve x^2+1 = 0 to get x = i or x = -i<BR>\n" );
document.write( "You could use the square root method.<BR>\n" );
document.write( "Or the quadratic formula could be applied (even if it may be a bit overkill).\r<BR>\n" );
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document.write( "The four roots of P(x) are:<BR>\n" );
document.write( "x = 3+3i<BR>\n" );
document.write( "x = 3-3i<BR>\n" );
document.write( "x = i<BR>\n" );
document.write( "x = -i<BR>\n" );
document.write( "where i = sqrt(-1)\r<BR>\n" );
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document.write( "Confirmation through use of WolframAlpha<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.wolframalpha.com/input/?i=x%5E4-6x%5E3%2B19x%5E2-6x%2B18%3D0\">https://www.wolframalpha.com/input/?i=x%5E4-6x%5E3%2B19x%5E2-6x%2B18%3D0</a>\r<BR>\n" );
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document.write( "GeoGebra also can be used to confirm the answer.<BR>\n" );
document.write( "You need to be in the CAS mode and you'd use the function called CSolve.<BR>\n" );
document.write( "The \"C\" in CSolve stands for \"complex\"<BR>\n" );
document.write( "The regular \"solve\" function will return an empty set to indicate \"no real solutions\", which is why you need to avoid using it.<BR>\n" );
document.write( "More info found here<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://wiki.geogebra.org/en/CSolve_Command\">https://wiki.geogebra.org/en/CSolve_Command</a><BR>\n" );
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