document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=94754>Question 94754</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor the polynomial completely.\r<BR>\n" );
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document.write( "I have factored other polynomials, but this one confuses me. The equation is...<BR>\n" );
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document.write( "-3s^2-10s+8\r<BR>\n" );
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document.write( "Thank you very much for your time! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #68997 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=68997\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3s%5E2-10s%2B8 ALIGN=MIDDLE  ALT=\"-3s%5E2-10s%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "To factor this, you make things a little easier by factoring out -1. If you do this the <BR>\n" );
document.write( "result is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%29%2A%283s%5E2+%2B+10s+-+8%29 ALIGN=MIDDLE  ALT=\"%28-1%29%2A%283s%5E2+%2B+10s+-+8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Let's not forget the -1 factor, but let's now concentrate on factoring the polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3s%5E2%2B10s-8 ALIGN=MIDDLE  ALT=\"3s%5E2%2B10s-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "Since the 3 preceding the s-squared term can only be factored into 3 and 1 we know if the<BR>\n" );
document.write( "polynomial can be factored, the factors must be of the form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283s+%2B-+a+%29%28s+%2B-+b+%29 ALIGN=MIDDLE  ALT=\"%283s+%2B-+a+%29%28s+%2B-+b+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "where \"a\" and \"b\" must be a factor pair of 8 ... in other words \"a\" and \"b\" must be either<BR>\n" );
document.write( "4 and 2 or 8 and 1.  And, considering the signs of \"a\" and \"b\", the terms must multiply with<BR>\n" );
document.write( "3 and 1 to result in the middle value + 10.  After some thought, you might see that if<BR>\n" );
document.write( "\"b\" were +4 and \"a\" were -2, the factors would become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283s+-2%29%28s+%2B+4%29 ALIGN=MIDDLE  ALT=\"%283s+-2%29%28s+%2B+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The cross product of +4 times +3s is +12s. And the cross product of -2 and s is -2s.  These<BR>\n" );
document.write( "cross products combine to give +12s - 2s = +10s which is the middle term of the polynomial<BR>\n" );
document.write( "being factored.<BR>\n" );
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document.write( "But don't forget that we originally factored a -1 from the polynomial. Therefore, when<BR>\n" );
document.write( "we put that factor into the solution the answer becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%29%2A%283s+-+2%29%28+s+%2B+4%29 ALIGN=MIDDLE  ALT=\"%28-1%29%2A%283s+-+2%29%28+s+%2B+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and by convention, the \"1\" is understood so the answer is just:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%283s-2%29%28s%2B4%29 ALIGN=MIDDLE  ALT=\"-%283s-2%29%28s%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is the answer. You could also move the minus sign into one set of parentheses. <BR>\n" );
document.write( "For example, you could have the answer in the form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283s-2%29%28-s-4%29 ALIGN=MIDDLE  ALT=\"%283s-2%29%28-s-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and if you multiply this out you will get back to the original polynomial you were given<BR>\n" );
document.write( "to factor.  <BR>\n" );
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document.write( "Hope that this helps you to understand the way to get to an answer for this problem.<BR>\n" );
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