document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198584>Question 1198584</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following polynomial equation  x^5-13x^3+36x=0\r<BR>\n" );
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document.write( "One of the following is the correct answer. Which one?\r<BR>\n" );
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document.write( "A) x=0, x=+or-2, x=+or-3\r<BR>\n" );
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document.write( "B) x=1, x=2, x=18\r<BR>\n" );
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document.write( "C) x=+or-2, x+or-3\r<BR>\n" );
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document.write( "D) x=+or-3, x=+or-5\r<BR>\n" );
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document.write( "E) x=0, x=+or-3, x=+or-12 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832440 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=832440\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "First let's pull out the GCF x<BR>\n" );
document.write( "x^5-13x^3+36x = 0<BR>\n" );
document.write( "x(x^4-13x^2+36) = 0\r<BR>\n" );
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document.write( "Now let's factor x^4-13x^2+36<BR>\n" );
document.write( "To do so, we have at least two methods\r<BR>\n" );
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document.write( "Method 1) <BR>\n" );
document.write( "Let w = x^2<BR>\n" );
document.write( "So w^2 = x^4\r<BR>\n" );
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document.write( "Then x^4-13x^2+36 is the same as w^2-13w+36<BR>\n" );
document.write( "Through trial and error, that would factor to (w-4)(w-9)<BR>\n" );
document.write( "Note: -4 and -9 add to -13 and multiply to 36.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "(w-4)(w-9) = (x^2-4)(x^2-9)<BR>\n" );
document.write( "(w-4)(w-9) = (x-2)(x+2)(x-3)(x+3)<BR>\n" );
document.write( "after applying the difference of squares rule.\r<BR>\n" );
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document.write( "So overall,<BR>\n" );
document.write( "x^5-13x^3+36x = 0<BR>\n" );
document.write( "x(x^4-13x^2+36) = 0<BR>\n" );
document.write( "x(x-2)(x+2)(x-3)(x+3) = 0<BR>\n" );
document.write( "To find the roots, set each factor equal to zero and solve for x.<BR>\n" );
document.write( "Example: x-2 = 0 leads to x = 2 as one root.\r<BR>\n" );
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document.write( "Therefore, the five roots are: <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%22%22+%2B-+2 ALIGN=MIDDLE  ALT=\"x+=+%22%22+%2B-+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%22%22+%2B-+3 ALIGN=MIDDLE  ALT=\"x+=+%22%22+%2B-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is the shorthand way of saying<BR>\n" );
document.write( "x = 0, x = -2, x = 2, x = -3, x = 3\r<BR>\n" );
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document.write( "Answer: Choice A\r<BR>\n" );
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document.write( "--------------------------------------------------------\r<BR>\n" );
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document.write( "Method 2)\r<BR>\n" );
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document.write( "Here's another way to factor x^4-13x^2+36\r<BR>\n" );
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document.write( "Use the rational root theorem to list out the factors of the last term 36: <BR>\n" );
document.write( "1, 2, 3, 4, 6, 9, 12, 18, 36<BR>\n" );
document.write( "List the negative factors as well<BR>\n" );
document.write( "-1, -2, -3, -4, -6, -9, -12, -18, -36\r<BR>\n" );
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document.write( "Then through trial and error, you should find that  x = -2, x = 2, x = -3, x = 3 are actual roots since they cause x^4-13x^2+36 to be zero.<BR>\n" );
document.write( "This gives the factorization (x-2)(x+2)(x-3)(x+3)\r<BR>\n" );
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document.write( "I don't recommend this second method as the first method is more efficient. But it's good to keep an alternative in mind.<BR>\n" );
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