document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=101585>Question 101585</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have tried this using our book and I just don't udestand the process. My teacher is of no help, she just keeps pointing me to our book.\r<BR>\n" );
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document.write( "Factor each polynomial completely.\r<BR>\n" );
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document.write( "a)   x^3y+2x^2y^2+xy^3\r<BR>\n" );
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document.write( "b)   -4w^3-16w^2+20w\r<BR>\n" );
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document.write( "c)   3x^2 – 17x + 10\r<BR>\n" );
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document.write( "Thank you so much for any help you can provide me with. :)\r<BR>\n" );
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document.write( "d)   -36a^2b + 21ab^2 – 3b^3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #73937 by </B><A HREF=/tutors/aboutme.mpl?userid=ptaylor><B>ptaylor(2198)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ptaylor><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/cgi-bin/show-face.mpl?user=ptaylor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=73937\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I have tried this using our book and I just don't udestand the process. My teacher is of no help, she just keeps pointing me to our book. <BR>\n" );
document.write( "Factor each polynomial completely. <BR>\n" );
document.write( "a) x^3y+2x^2y^2+xy^3 <BR>\n" );
document.write( "b) -4w^3-16w^2+20w <BR>\n" );
document.write( "c) 3x^2 – 17x + 10 <BR>\n" );
document.write( "Thank you so much for any help you can provide me with. :) <BR>\n" );
document.write( "d) -36a^2b + 21ab^2 – 3b^3\r<BR>\n" );
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document.write( "a) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3y%2B2x%5E2y%5E2%2Bxy%5E3 ALIGN=MIDDLE  ALT=\"x%5E3y%2B2x%5E2y%5E2%2Bxy%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > by inspection, we see that we can take an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy ALIGN=MIDDLE  ALT=\"xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of each term so lets do that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy%28x%5E2%2B2xy%2By%5E2%29 ALIGN=MIDDLE  ALT=\"xy%28x%5E2%2B2xy%2By%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Again, by inspection, we see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2xy%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B2xy%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a perfect square<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2By%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2By%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  but we can also do the following:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy%28x%5E2%2Bxy%2Bxy%2By%5E2%29 ALIGN=MIDDLE  ALT=\"xy%28x%5E2%2Bxy%2Bxy%2By%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we can re-write this as follows:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy%28x%28x%2By%29%2By%28x%2By%29%29 ALIGN=MIDDLE  ALT=\"xy%28x%28x%2By%29%2By%28x%2By%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > now we can factor out an<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2By%29 ALIGN=MIDDLE  ALT=\"%28x%2By%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy%28x%2By%29%28x%2By%29 ALIGN=MIDDLE  ALT=\"xy%28x%2By%29%28x%2By%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy%28x%2By%29%5E2 ALIGN=MIDDLE  ALT=\"xy%28x%2By%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >------------------ans\r<BR>\n" );
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document.write( "b) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4w%5E3-16w%5E2%2B20w+ ALIGN=MIDDLE  ALT=\"-4w%5E3-16w%5E2%2B20w+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  by inspection, we see that we can take a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4w ALIGN=MIDDLE  ALT=\"-4w\"   DISABLED_style= \"max-width:90%; height:auto;\" >out of each term so lets do that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4w%28w%5E2%2B4w-5%29 ALIGN=MIDDLE  ALT=\"-4w%28w%5E2%2B4w-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Now we observe that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5E2%2B4w-5 ALIGN=MIDDLE  ALT=\"w%5E2%2B4w-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a quadratic in standard form.  When we have a quadratic in standard form and the A coefficient is 1 then the B coefficient is the sum of the factors of the C coefficent.  What are the factors of the C coefficient??  They can only be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-5+and+%2B1%29 ALIGN=MIDDLE  ALT=\"%28-5+and+%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%2B5+and+-1%29 ALIGN=MIDDLE  ALT=\"%28%2B5+and+-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  By inspection we see that +5 and -1=+4.  So now we know that: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5E2%2B4w-5 ALIGN=MIDDLE  ALT=\"w%5E2%2B4w-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28w%2B5%29%28w-1%29 ALIGN=MIDDLE  ALT=\"%28w%2B5%29%28w-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now putting it all back together, we have:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4w%28w-5%29%28w%2B1%29 ALIGN=MIDDLE  ALT=\"-4w%28w-5%29%28w%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---------------------ans \r<BR>\n" );
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document.write( "c) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+%96+17x+%2B+10 ALIGN=MIDDLE  ALT=\"3x%5E2+%96+17x+%2B+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Here are the possibilities:<BR>\n" );
document.write( "By inspection, we see that if this quadratic can be factored, then the factors must be of the form (a-b)(c-d) or (c-b)(a-d).  Why???  Because the last term is positive and the middle term is negative.  So here are the possibilities:<BR>\n" );
document.write( "1.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x-10%29%28x-1%29 ALIGN=MIDDLE  ALT=\"%283x-10%29%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-10%29%283x-1%29 ALIGN=MIDDLE  ALT=\"%28x-10%29%283x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "3.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x-5%29%28x-2%29 ALIGN=MIDDLE  ALT=\"%283x-5%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "4.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5%29%283x-2%29 ALIGN=MIDDLE  ALT=\"%28x-5%29%283x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Expanding each of the above using the FOIL crutch (First, Inner, Outer, Last) we get:\r<BR>\n" );
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document.write( "1.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-10x-3x%2B10 ALIGN=MIDDLE  ALT=\"3x%5E2-10x-3x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >-----------------NO!!! inner term is -13x<BR>\n" );
document.write( "2.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-30x-x%2B10 ALIGN=MIDDLE  ALT=\"3x%5E2-30x-x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >------------------NO!!! inner term is -31x<BR>\n" );
document.write( "3.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-5x-2x%2B10 ALIGN=MIDDLE  ALT=\"3x%5E2-5x-2x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >-------------------NO!!!! inner term is -7x<BR>\n" );
document.write( "4.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-15x-2x%2B10 ALIGN=MIDDLE  ALT=\"3x%5E2-15x-2x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >----------------BINGO!!!!!!!\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5%29%283x-2%29 ALIGN=MIDDLE  ALT=\"%28x-5%29%283x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >-----------------------ans\r<BR>\n" );
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document.write( "d) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-36a%5E2b%2B21ab%5E2-3b%5E3 ALIGN=MIDDLE  ALT=\"-36a%5E2b%2B21ab%5E2-3b%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >   By inspection, we see that we can take a -3b out of each term:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3b%2812a%5E2-7ab%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"-3b%2812a%5E2-7ab%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now use the approach that was used in (c) above and find the factors for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12a%5E2-7ab%2Bb%5E2 ALIGN=MIDDLE  ALT=\"12a%5E2-7ab%2Bb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .  I bet you can do it!!!!!\r<BR>\n" );
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document.write( "Hope this helps---ptaylor\r<BR>\n" );
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document.write( " </SPAN>\n" );
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