document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=925070>Question 925070</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1) List all of the possible roots of : 2x^3 - 5x^2 - 28x + 15 = 0. <BR>\n" );
document.write( "2) Then find all the roots. \r<BR>\n" );
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document.write( "I'm confused because I thought possible roots and all roots were the same thing? Also, are \"zeros\" also roots? Thank you.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #561288 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=561288\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Yes, roots and zeros are synonymous. I prefer the term \"root\" since zero is a number and it gets muddied up a bit.\r<BR>\n" );
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document.write( "1) The possible rational roots of f(x) are potential values that could make f(x) = 0 true.\r<BR>\n" );
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document.write( "In general, not all possible roots are actual roots. You will have more possible roots than actual roots for the majority of the cases.\r<BR>\n" );
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document.write( "To list the possible rational roots, you use the <a  rel=nofollow HREF=\"http://www.mathwords.com/r/rational_root_theorem.htm\">rational root theorem</a>. The idea is to list the factors of the first and last coefficients, then divide each factor\r<BR>\n" );
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document.write( "Factors of 15: 1, 3, 5, 15<BR>\n" );
document.write( "Factor of 2: 1, 2\r<BR>\n" );
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document.write( "Divide each factor of 15 by each factor of 2 to get these fractions: 1/1, 1/2, 3/1, 3/2, 5/1, 5/2, 15/1, 15/2\r<BR>\n" );
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document.write( "Those fractions simplify to 1, 1/2, 3, 3/2, 5, 5/2, 15, 15/2\r<BR>\n" );
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document.write( "Now add on the negative counterparts: -1, -1/2, -3, -3/2, -5, -5/2, -15, -15/2\r<BR>\n" );
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document.write( "So the list of possible rational roots is shown below in <font color=\"blue\">blue</font>\r<BR>\n" );
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document.write( "1, 1/2, 3, 3/2, 5, 5/2, 15, 15/2<BR>\n" );
document.write( "-1, -1/2, -3, -3/2, -5, -5/2, -15, -15/2<BR>\n" );
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document.write( "2) To find the actual rational roots, you need to check every possible rational root. To check each one, you plug each one (one at a time) into f(x). \r<BR>\n" );
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document.write( "If you get f(x) = 0, then that possible root is indeed an actual root.\r<BR>\n" );
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document.write( "In this case, f(x) = 2x^3 - 5x^2 - 28x + 15\r<BR>\n" );
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document.write( "I'll do the first possible root\r<BR>\n" );
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document.write( "Checking x = 1\r<BR>\n" );
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document.write( "f(x) = 2x^3 - 5x^2 - 28x + 15\r<BR>\n" );
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document.write( "f(1) = 2(1)^3 - 5(1)^2 - 28(1) + 15\r<BR>\n" );
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document.write( "f(1) = -16\r<BR>\n" );
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document.write( "The result is -16 and NOT 0. So x = 1 is NOT an actual root.\r<BR>\n" );
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document.write( "I'll let you check the other potential rational roots.\r<BR>\n" );
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document.write( "Let me know if that helps or not. Thanks.\r<BR>\n" );
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document.write( "If you need more help, feel free to email me at <a  rel=nofollow HREF=\"mailto:jim_thompson5910@hotmail.com?Subject=I%20Need%20Algebra%20Help\">jim_thompson5910@hotmail.com</a>\r<BR>\n" );
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