document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=139593>Question 139593</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0618250182/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Factor each expression.\r<BR>\n" );
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document.write( "5x2+16x+3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #101782 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=101782\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2%2B16x%2B3 ALIGN=MIDDLE  ALT=\"5x%5E2%2B16x%2B3\"   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=5x%5E2 ALIGN=MIDDLE  ALT=\"5x%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=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 5 and 3 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 5 and the last coefficient 3 to get 15. Now what two numbers multiply to 15 and add to the  middle coefficient 16? Let's list all of the factors of 15:\r<BR>\n" );
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document.write( "Factors of 15:\r<BR>\n" );
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document.write( "1,3,5,15\r<BR>\n" );
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document.write( "-1,-3,-5,-15 ...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 15\r<BR>\n" );
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document.write( "1*15\r<BR>\n" );
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document.write( "3*5\r<BR>\n" );
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document.write( "(-1)*(-15)\r<BR>\n" );
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document.write( "(-3)*(-5)\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 16? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 16\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\">15</TD><TD >1+15=16</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">5</TD><TD >3+5=8</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-15</TD><TD >-1+(-15)=-16</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">-5</TD><TD >-3+(-5)=-8</TD></tr>\n" );
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document.write( "From this list we can see that 1 and 15 add up to 16 and multiply to 15\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=5x%5E2%2B16x%2B3 ALIGN=MIDDLE  ALT=\"5x%5E2%2B16x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x ALIGN=MIDDLE  ALT=\"16x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%2B15x ALIGN=MIDDLE  ALT=\"1x%2B15x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%2B15x ALIGN=MIDDLE  ALT=\"1x%2B15x\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x ALIGN=MIDDLE  ALT=\"16x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x ALIGN=MIDDLE  ALT=\"16x\"   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=5x%5E2%2Bhighlight%281x%2B15x%29%2B3 ALIGN=MIDDLE  ALT=\"5x%5E2%2Bhighlight%281x%2B15x%29%2B3\"   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=5x%5E2%2B1x%2B15x%2B3 ALIGN=MIDDLE  ALT=\"5x%5E2%2B1x%2B15x%2B3\"   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=%285x%5E2%2B1x%29%2B%2815x%2B3%29 ALIGN=MIDDLE  ALT=\"%285x%5E2%2B1x%29%2B%2815x%2B3%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%285x%2B1%29%2B3%285x%2B1%29 ALIGN=MIDDLE  ALT=\"x%285x%2B1%29%2B3%285x%2B1%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=3 ALIGN=MIDDLE  ALT=\"3\"   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%2B3%29%285x%2B1%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%285x%2B1%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=5x%2B1 ALIGN=MIDDLE  ALT=\"5x%2B1\"   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=5x%5E2%2B1x%2B15x%2B3 ALIGN=MIDDLE  ALT=\"5x%5E2%2B1x%2B15x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%285x%2B1%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%285x%2B1%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=5x%5E2%2B16x%2B3 ALIGN=MIDDLE  ALT=\"5x%5E2%2B16x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%285x%2B1%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%285x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2%2B16x%2B3 ALIGN=MIDDLE  ALT=\"5x%5E2%2B16x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2%2B1x%2B15x%2B3 ALIGN=MIDDLE  ALT=\"5x%5E2%2B1x%2B15x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "-------------------------------\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=5x%5E2%2B16x%2B3 ALIGN=MIDDLE  ALT=\"5x%5E2%2B16x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%285x%2B1%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%285x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( " </SPAN>\n" );
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