document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=674305>Question 674305</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can someone walk me through the steps for the following problem? Thank you! \r<BR>\n" );
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document.write( "x^2-3xy+9y^2\r<BR>\n" );
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document.write( "1. Factor completely. <BR>\n" );
document.write( "2. Remember to look first for a common factor. <BR>\n" );
document.write( "3. Check by multiplying. <BR>\n" );
document.write( "4. If a polynomial is prime, state this. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #419145 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=419145\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Looking at the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-3xy%2B9y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-3xy%2B9y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see that the first coefficient is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the second coefficient is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the last coefficient is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now multiply the first coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the last coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%289%29=9 ALIGN=MIDDLE  ALT=\"%281%29%289%29=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now the question is: what two whole numbers multiply to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the previous product) <font size=4><b>and</b></font> add to the second coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >?\r<BR>\n" );
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document.write( "To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the previous product).\r<BR>\n" );
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document.write( "Factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "1,3,9\r<BR>\n" );
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document.write( "-1,-3,-9\r<BR>\n" );
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document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r<BR>\n" );
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document.write( "These factors pair up and multiply to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "1*9 = 9<BR>\n" );
document.write( "3*3 = 9<BR>\n" );
document.write( "(-1)*(-9) = 9<BR>\n" );
document.write( "(-3)*(-3) = 9\r<BR>\n" );
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document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<th>First Number</th><th>Second Number</th><th>Sum</th><tr><TD   align=\"center\"><font color=black>1</font></TD><TD   align=\"center\"><font color=black>9</font></TD><TD   align=\"center\"><font color=black>1+9=10</font></TD></tr><tr><TD   align=\"center\"><font color=black>3</font></TD><TD   align=\"center\"><font color=black>3</font></TD><TD   align=\"center\"><font color=black>3+3=6</font></TD></tr><tr><TD   align=\"center\"><font color=black>-1</font></TD><TD   align=\"center\"><font color=black>-9</font></TD><TD   align=\"center\"><font color=black>-1+(-9)=-10</font></TD></tr><tr><TD   align=\"center\"><font color=black>-3</font></TD><TD   align=\"center\"><font color=black>-3</font></TD><TD   align=\"center\"><font color=black>-3+(-3)=-6</font></TD></tr>\n" );
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document.write( "From the table, we can see that there are no pairs of numbers which add to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-3xy%2B9y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-3xy%2B9y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > cannot be factored.\r<BR>\n" );
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document.write( "<a name=\"ans\">\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-3xy%2B9y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-3xy%2B9y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > doesn't factor at all (over the rational numbers).\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-3xy%2B9y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-3xy%2B9y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is prime.<BR>\n" );
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