document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=465494>Question 465494</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the polynomial f(x) = 2x4 - 18x2	\r<BR>\n" );
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document.write( "a.	Use Descartes Rule of Signs to determine the number of positive and negative roots.<BR>\n" );
document.write( "b.	Use the Rational Zero Theorem (aka Rational Roots Theorem) to determine a list of possible zeros.<BR>\n" );
document.write( "c.	Use the Intermediate Value Theorem to prove that the polynomial has a zero in the interval [-6,-1].<BR>\n" );
document.write( "d.	Solve for the zeros of f(x).\r<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 #318994 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=318994\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+2x%5E4+-+18x%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29+=+2x%5E4+-+18x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If x = 0 (up to multiplicity) is an obvious root of the polynomial, then factor out the highest power of x from the expression, and then apply the relevant theorem.  Hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+2x%5E2%28x%5E2+-+9%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+2x%5E2%28x%5E2+-+9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and apply simply look at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+9 ALIGN=MIDDLE  ALT=\"x%5E2+-+9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "a. There is only one variation of sign among the terms of the polynomial, and so there is one positive real root.  If we substitute -x for x in the polynomial, we get the same function, and so this tells us that there is also one negative root.\r<BR>\n" );
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document.write( "b.  From the rational roots theorem, the possible rational roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+9 ALIGN=MIDDLE  ALT=\"x%5E2+-+9\"   DISABLED_style= \"max-width:90%; height:auto;\" > are simply the divisors of 9, namely -3, -1,1, and 3.\r<BR>\n" );
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document.write( "c. Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+9 ALIGN=MIDDLE  ALT=\"x%5E2+-+9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%29%5E2+-+9+=+1-9+=+-8+%3C+0 ALIGN=MIDDLE  ALT=\"%28-1%29%5E2+-+9+=+1-9+=+-8+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, while <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-6%29%5E2+-+9+=+36-9+=+27+%3E0 ALIGN=MIDDLE  ALT=\"%28-6%29%5E2+-+9+=+36-9+=+27+%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, hence by the intermediate value theorem, there is r between (-6, -1) such that f(r) = 0.\r<BR>\n" );
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document.write( "d.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+2x%5E4+-+18x%5E2+=+2x%5E2%28x%5E2+-+9%29+=+2x%5E2%28x-3%29%28x%2B3%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+2x%5E4+-+18x%5E2+=+2x%5E2%28x%5E2+-+9%29+=+2x%5E2%28x-3%29%28x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> x = 0,0,-3,3 are the roots.\r<BR>\n" );
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