document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=117516>Question 117516</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" src=\"http://img338.imageshack.us/img338/9167/mathaq6.png\"><BR>\n" );
document.write( "I NEED HELP! :] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #85518 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=85518\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc%2Bd=0 ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc%2Bd=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=-a-b-c ALIGN=MIDDLE  ALT=\"d=-a-b-c\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve for d\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E3%2Bbx%5E2%2Bcx-a-b-c=0 ALIGN=MIDDLE  ALT=\"ax%5E3%2Bbx%5E2%2Bcx-a-b-c=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=-a-b-c ALIGN=MIDDLE  ALT=\"d=-a-b-c\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E3%2Bbx%5E2%2Bcx%2Bd=0 ALIGN=MIDDLE  ALT=\"ax%5E3%2Bbx%5E2%2Bcx%2Bd=0\"   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=%28ax%5E3-a%29%2B%28bx%5E2-b%29%2B%28cx-c%29=0 ALIGN=MIDDLE  ALT=\"%28ax%5E3-a%29%2B%28bx%5E2-b%29%2B%28cx-c%29=0\"   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=a%28x%5E3-1%29%2Bb%28x%5E2-1%29%2Bc%28x-1%29=0 ALIGN=MIDDLE  ALT=\"a%28x%5E3-1%29%2Bb%28x%5E2-1%29%2Bc%28x-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF from each individual group\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x-1%29%28x%5E2%2Bx%2B1%29%2Bb%28x%5E2-1%29%2Bc%28x-1%29=0 ALIGN=MIDDLE  ALT=\"a%28x-1%29%28x%5E2%2Bx%2B1%29%2Bb%28x%5E2-1%29%2Bc%28x-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-1 ALIGN=MIDDLE  ALT=\"x%5E3-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > by using the difference of cubes formula to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%28x%5E2%2Bx%2B1%29 ALIGN=MIDDLE  ALT=\"%28x-1%29%28x%5E2%2Bx%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/plot-formula.mpl?expression=a%28x-1%29%28x%5E2%2Bx%2B1%29%2Bb%28x-1%29%28x%2B1%29%2Bc%28x-1%29=0 ALIGN=MIDDLE  ALT=\"a%28x-1%29%28x%5E2%2Bx%2B1%29%2Bb%28x-1%29%28x%2B1%29%2Bc%28x-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-1 ALIGN=MIDDLE  ALT=\"x%5E2-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > by using the difference of squares formula to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%28x%2B1%29 ALIGN=MIDDLE  ALT=\"%28x-1%29%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice how we have the common term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1 ALIGN=MIDDLE  ALT=\"x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We can factor this term out.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%28a%28x%5E2%2Bx%2B1%29%2Bb%28x%2B1%29%2Bc%29=0 ALIGN=MIDDLE  ALT=\"%28x-1%29%28a%28x%5E2%2Bx%2B1%29%2Bb%28x%2B1%29%2Bc%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1 ALIGN=MIDDLE  ALT=\"x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now if we let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H=x-1 ALIGN=MIDDLE  ALT=\"H=x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=K=a%28x%5E2%2Bx%2B1%29%2Bb%28x%2B1%29%2Bc ALIGN=MIDDLE  ALT=\"K=a%28x%5E2%2Bx%2B1%29%2Bb%28x%2B1%29%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > we'll get\r<BR>\n" );
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document.write( "So by the zero product property we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H=0 ALIGN=MIDDLE  ALT=\"H=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=K=0 ALIGN=MIDDLE  ALT=\"K=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "but since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H=x-1 ALIGN=MIDDLE  ALT=\"H=x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means\r<BR>\n" );
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document.write( "Now solve for x\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E3%2Bbx%5E2%2Bcx%2Bd=0 ALIGN=MIDDLE  ALT=\"ax%5E3%2Bbx%5E2%2Bcx%2Bd=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > if the coefficients satisfy the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc%2Bd=0 ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc%2Bd=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "Unfortunately, there is no formal check for this type of problem. But if we make sure that the coefficients satisfy the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc%2Bd=0 ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc%2Bd=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then we can graph the equation and see that one root is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. For instance, let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=b=1 ALIGN=MIDDLE  ALT=\"a=b=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=d=-1 ALIGN=MIDDLE  ALT=\"c=d=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > so we'll get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2Bx%5E2-x-1 ALIGN=MIDDLE  ALT=\"x%5E3%2Bx%5E2-x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice how <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B1-1-1=2-2=0 ALIGN=MIDDLE  ALT=\"1%2B1-1-1=2-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > which satisfies the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc%2Bd=0 ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc%2Bd=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Now graph the polynomial to get\r<BR>\n" );
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document.write( "and we can see that one root is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If we try different values of a,b,c, and d that will satisfy the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc%2Bd=0 ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc%2Bd=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we'll see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a root every time. So in a way, our answer has been verified. </SPAN>\n" );
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