document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=118105>Question 118105</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need to learn how to factor completly\r<BR>\n" );
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document.write( "3x^2 + 17x + 10 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #86208 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=86208\">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 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B17x%2B10 ALIGN=MIDDLE  ALT=\"3x%5E2%2B17x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that the first term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2 ALIGN=MIDDLE  ALT=\"3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the last term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 3 and 10 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 3 and the last coefficient 10 to get 30. Now what two numbers multiply to 30 and add to the  middle coefficient 17? Let's list all of the factors of 30:\r<BR>\n" );
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document.write( "Factors of 30:\r<BR>\n" );
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document.write( "1,2,3,5,6,10,15,30\r<BR>\n" );
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document.write( "-1,-2,-3,-5,-6,-10,-15,-30 ...List the negative factors as well. 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 30\r<BR>\n" );
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document.write( "1*30\r<BR>\n" );
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document.write( "2*15\r<BR>\n" );
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document.write( "3*10\r<BR>\n" );
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document.write( "5*6\r<BR>\n" );
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document.write( "(-1)*(-30)\r<BR>\n" );
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document.write( "(-2)*(-15)\r<BR>\n" );
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document.write( "(-3)*(-10)\r<BR>\n" );
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document.write( "(-5)*(-6)\r<BR>\n" );
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document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
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document.write( "Now which of these pairs add to 17? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 17\r<BR>\n" );
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document.write( "<th>First Number</th><th>Second Number</th><th>Sum</th><tr><TD  align=\"center\">1</TD><TD  align=\"center\">30</TD><TD >1+30=31</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">15</TD><TD >2+15=17</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">10</TD><TD >3+10=13</TD></tr><tr><TD  align=\"center\">5</TD><TD  align=\"center\">6</TD><TD >5+6=11</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-30</TD><TD >-1+(-30)=-31</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">-15</TD><TD >-2+(-15)=-17</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">-10</TD><TD >-3+(-10)=-13</TD></tr><tr><TD  align=\"center\">-5</TD><TD  align=\"center\">-6</TD><TD >-5+(-6)=-11</TD></tr>\n" );
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document.write( "From this list we can see that 2 and 15 add up to 17 and multiply to 30\r<BR>\n" );
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document.write( "Now looking at the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B17x%2B10 ALIGN=MIDDLE  ALT=\"3x%5E2%2B17x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17x ALIGN=MIDDLE  ALT=\"17x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B15x ALIGN=MIDDLE  ALT=\"2x%2B15x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B15x ALIGN=MIDDLE  ALT=\"2x%2B15x\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17x ALIGN=MIDDLE  ALT=\"17x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17x ALIGN=MIDDLE  ALT=\"17x\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2Bhighlight%282x%2B15x%29%2B10 ALIGN=MIDDLE  ALT=\"3x%5E2%2Bhighlight%282x%2B15x%29%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B2x%2B15x%2B10 ALIGN=MIDDLE  ALT=\"3x%5E2%2B2x%2B15x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" > by grouping:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%5E2%2B2x%29%2B%2815x%2B10%29 ALIGN=MIDDLE  ALT=\"%283x%5E2%2B2x%29%2B%2815x%2B10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%283x%2B2%29%2B5%283x%2B2%29 ALIGN=MIDDLE  ALT=\"x%283x%2B2%29%2B5%283x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the first group. Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the second group\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B5%29%283x%2B2%29 ALIGN=MIDDLE  ALT=\"%28x%2B5%29%283x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since we have a common term of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B2 ALIGN=MIDDLE  ALT=\"3x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can combine like terms\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B2x%2B15x%2B10 ALIGN=MIDDLE  ALT=\"3x%5E2%2B2x%2B15x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B5%29%283x%2B2%29 ALIGN=MIDDLE  ALT=\"%28x%2B5%29%283x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So this also means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B17x%2B10 ALIGN=MIDDLE  ALT=\"3x%5E2%2B17x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B5%29%283x%2B2%29 ALIGN=MIDDLE  ALT=\"%28x%2B5%29%283x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B17x%2B10 ALIGN=MIDDLE  ALT=\"3x%5E2%2B17x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B2x%2B15x%2B10 ALIGN=MIDDLE  ALT=\"3x%5E2%2B2x%2B15x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\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=3x%5E2%2B17x%2B10 ALIGN=MIDDLE  ALT=\"3x%5E2%2B17x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B5%29%283x%2B2%29 ALIGN=MIDDLE  ALT=\"%28x%2B5%29%283x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );