document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=148119>Question 148119</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor: (2x^(2/3)) - (5x^(1/3)) - 3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #108504 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=108504\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2x^{\frac{2}{3}} - 5x^{\frac{1}{3}} - 3\"> Start with the given expression.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE z=x^{\frac{1}{3}}\">. This means that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE z^{2}=\left(x^{\frac{1}{3}}\right)^{2}=x^{\frac{2}{3}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2z^{2} - 5z - 3\"> Replace <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^{\frac{2}{3}}\"> with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE z^{2}\">. Replace <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^{\frac{1}{3}}\"> with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE z\">\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z%5E2-5z-3 ALIGN=MIDDLE  ALT=\"2z%5E2-5z-3\"   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=2z%5E2 ALIGN=MIDDLE  ALT=\"2z%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 2 and -3 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 2 and the last coefficient -3 to get -6. Now what two numbers multiply to -6 and add to the  middle coefficient -5? Let's list all of the factors of -6:\r<BR>\n" );
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document.write( "Factors of -6:\r<BR>\n" );
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document.write( "1,2,3,6\r<BR>\n" );
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document.write( "-1,-2,-3,-6 ...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 -6\r<BR>\n" );
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document.write( "(1)*(-6)\r<BR>\n" );
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document.write( "(2)*(-3)\r<BR>\n" );
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document.write( "(-1)*(6)\r<BR>\n" );
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document.write( "(-2)*(3)\r<BR>\n" );
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document.write( "note: remember, the product of a negative and a positive number is a negative number\r<BR>\n" );
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document.write( "Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5\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\">-6</TD><TD >1+(-6)=-5</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">-3</TD><TD >2+(-3)=-1</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">6</TD><TD >-1+6=5</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">3</TD><TD >-2+3=1</TD></tr>\n" );
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document.write( "From this list we can see that 1 and -6 add up to -5 and multiply to -6\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=2z%5E2-5z-3 ALIGN=MIDDLE  ALT=\"2z%5E2-5z-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5z ALIGN=MIDDLE  ALT=\"-5z\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1z%2B-6z ALIGN=MIDDLE  ALT=\"1z%2B-6z\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1z%2B-6z ALIGN=MIDDLE  ALT=\"1z%2B-6z\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5z ALIGN=MIDDLE  ALT=\"-5z\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5z ALIGN=MIDDLE  ALT=\"-5z\"   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=2z%5E2%2Bhighlight%281z%2B-6z%29%2B-3 ALIGN=MIDDLE  ALT=\"2z%5E2%2Bhighlight%281z%2B-6z%29%2B-3\"   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=2z%5E2%2Bz-6z-3 ALIGN=MIDDLE  ALT=\"2z%5E2%2Bz-6z-3\"   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=%282z%5E2%2Bz%29%2B%28-6z-3%29 ALIGN=MIDDLE  ALT=\"%282z%5E2%2Bz%29%2B%28-6z-3%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=z%282z%2B1%29-3%282z%2B1%29 ALIGN=MIDDLE  ALT=\"z%282z%2B1%29-3%282z%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=z ALIGN=MIDDLE  ALT=\"z\"   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=%28z-3%29%282z%2B1%29 ALIGN=MIDDLE  ALT=\"%28z-3%29%282z%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=2z%2B1 ALIGN=MIDDLE  ALT=\"2z%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=2z%5E2-5z-3 ALIGN=MIDDLE  ALT=\"2z%5E2-5z-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28z-3%29%282z%2B1%29 ALIGN=MIDDLE  ALT=\"%28z-3%29%282z%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(x^{\frac{1}{3}}-3\right)\left(2x^{\frac{1}{3}}+1\right)\"> Now replace \"z\" with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^{\frac{1}{3}}\"> \r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2x^{\frac{2}{3}} - 5x^{\frac{1}{3}} - 3\"> factors to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(x^{\frac{1}{3}}-3\right)\left(2x^{\frac{1}{3}}+1\right)\">\r<BR>\n" );
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document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2x^{\frac{2}{3}} - 5x^{\frac{1}{3}} - 3=\left(x^{\frac{1}{3}}-3\right)\left(2x^{\frac{1}{3}}+1\right)\"> where every variable is positive. </SPAN>\n" );
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