document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1037258>Question 1037258</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Check whether x+&#8730;3 is a factor of x^4+x^2-12<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #651947 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=651947\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you can use synthetic division to prove that it is a factor.\r<BR>\n" );
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document.write( "take x^4 + x^2 - 12 and fill in the missing terms in descending order of degree to get:\r<BR>\n" );
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document.write( "x^4 + 0x^3 + x^2 + 0x - 12.\r<BR>\n" );
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document.write( "take the coefficients of each of these terms to get:\r<BR>\n" );
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document.write( "1 + 0 + 1 + 0 - 12\r<BR>\n" );
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document.write( "take the factor of x + sqrt(3) and set it equal to 0 and solve for x to get:\r<BR>\n" );
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document.write( "x = -sqrt(3).\r<BR>\n" );
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document.write( "your synthetic division will be -sqrt(3) being synthetically divided into 1 + 0 + 1 + 0 - 12.\r<BR>\n" );
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document.write( "see the following worksheet for the details of the calculations.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/060601.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "if you don't know how to do synthetic division, then check out the following tutorial.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.purplemath.com/modules/synthdiv.htm\" target = \"_blank\">http://www.purplemath.com/modules/synthdiv.htm</a>\r<BR>\n" );
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document.write( "you could also have done the following:\r<BR>\n" );
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document.write( "solve for x + sqrt(3) = 0 to get x = -sqrt(3).\r<BR>\n" );
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document.write( "since f(x) = x^4+x^2-12, then replace x with -sqrt(3) to get:\r<BR>\n" );
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document.write( "f(-sqrt(3)) = (-sqrt(3))^4 + (-sqrt(3))^2 - 12.\r<BR>\n" );
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document.write( "evaluate this equation to get f(-sqrt(3)) = 0.\r<BR>\n" );
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document.write( "if f(a) is equal to 0, then a is a root and (x-a) is a factor.\r<BR>\n" );
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document.write( "you can also solve this graphically by graphing the equation of y = x^4 + x^2 - 12.\r<BR>\n" );
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document.write( "your graph will look like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/060602.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the graph shows that the roots of the equation are plus and minus 1.732.\r<BR>\n" );
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document.write( "sqrt(3) is equal to 1.732050808.\r<BR>\n" );
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document.write( "plus and minus 1.732 are just rounded versions of plus and minus sqrt(3).\r<BR>\n" );
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document.write( "the factors of x^4 + x^2 - 12 are (x^2 + 4) * (x^2 - 3).\r<BR>\n" );
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document.write( "set these factors to 0 and solve for x to get:\r<BR>\n" );
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document.write( "x^2 + 4 = 0<BR>\n" );
document.write( "subtract 4 from each side of the equation to get x^2 = -4<BR>\n" );
document.write( "take the square root of both sides of this equation to get x = plus or minus sqrt(-4).<BR>\n" );
document.write( "those are imaginary roots (also called complex), and therefore don't show up on the graph.\r<BR>\n" );
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document.write( "x^2 - 3 = 0<BR>\n" );
document.write( "add 3 to both sides of the equation to get x^2 = 3<BR>\n" );
document.write( "take the square root of both sides of this equation to get x = plus or minus sqrt(3).<BR>\n" );
document.write( "those are real roots, and therefore do show up on the graph.\r<BR>\n" );
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document.write( "if you look at the factor of (x^2 - 3), this can be simplified further into (x + sqrt(3)) * (x - sqrt(3)).\r<BR>\n" );
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document.write( "that's where the factor of (x + sqrt(3)) came from.\r<BR>\n" );
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