document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=541285>Question 541285</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor the expression completely. (This type of expression arises in calculus in using the \"product rule.\")\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+%284x-12%29%5E2+%2B+x%5E3+%2A+2%284x-12%29%2A4 ALIGN=MIDDLE  ALT=\"3x%5E2+%284x-12%29%5E2+%2B+x%5E3+%2A+2%284x-12%29%2A4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The final answer is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2%28x-3%29%285x-9%29 ALIGN=MIDDLE  ALT=\"16x%5E2%28x-3%29%285x-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Please show detailed step to obtain the answer so i can follow what you are doing. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #354151 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=354151\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given to factor:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+%284x-12%29%5E2+%2B+x%5E3+%2A+2%2A%284x-12%29%2A4 ALIGN=MIDDLE  ALT=\"3x%5E2+%284x-12%29%5E2+%2B+x%5E3+%2A+2%2A%284x-12%29%2A4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "I'm not sure that this is the most efficient way to get to the answer, but here's what I did:<BR>\n" );
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document.write( "I started by breaking down the exponent terms into their factors:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Ax%2Ax%2A%284x-12%29%2A%284x-12%29+%2B+x%2Ax%2Ax%2A2%2A%284x+-12%29+%2A4 ALIGN=MIDDLE  ALT=\"3%2Ax%2Ax%2A%284x-12%29%2A%284x-12%29+%2B+x%2Ax%2Ax%2A2%2A%284x+-12%29+%2A4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next I noted that the term (4x - 12) can be factored to 4*(x - 3). So I replaced each (4x - 12) with 4*(x - 3) to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Ax%2Ax%2A4%2A%28x+-+3%29%2A4%2A%28x+-3%29+%2B+x%2Ax%2Ax%2A2%2A4%2A%28x+-+3%29%2A4 ALIGN=MIDDLE  ALT=\"3%2Ax%2Ax%2A4%2A%28x+-+3%29%2A4%2A%28x+-3%29+%2B+x%2Ax%2Ax%2A2%2A4%2A%28x+-+3%29%2A4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then I pulled x*x from each of the two terms (the two terms are separated by the + sign):<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Ax%283%2A4%2A%28x-3%29%2A4%2A%28x-3%29%2B%28x%2A2%2A4%2A%28x-3%29%2A4%29%29 ALIGN=MIDDLE  ALT=\"x%2Ax%283%2A4%2A%28x-3%29%2A4%2A%28x-3%29%2B%28x%2A2%2A4%2A%28x-3%29%2A4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next I pulled out an (x-3) factor from both the terms:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Ax%2A%28x+-3%29%283%2A4%2A4%2A%28x-3%29%2Bx%2A2%2A4%2A4%29 ALIGN=MIDDLE  ALT=\"x%2Ax%2A%28x+-3%29%283%2A4%2A4%2A%28x-3%29%2Bx%2A2%2A4%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then I multiplied the constants. First to find that 3*4*4 = 48. Then to find that 2*4*4 = 32. This made the resulting expression become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Ax%2A%28x-3%29%2A%2848%2A%28x-3%29%2B+32%2Ax%29 ALIGN=MIDDLE  ALT=\"x%2Ax%2A%28x-3%29%2A%2848%2A%28x-3%29%2B+32%2Ax%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then I factored 16 out of each of the two terms in parentheses to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Ax%2A%28x-3%29%2A16%283%2A%28x-3%29%2B+2x%29 ALIGN=MIDDLE  ALT=\"x%2Ax%2A%28x-3%29%2A16%283%2A%28x-3%29%2B+2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next I multiplied out the 3 times (x-3) to get 3x - 9 inside the parentheses and get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Ax%2A%28x-3%29%2A16%2A%283x+-+9+%2B+2x%29 ALIGN=MIDDLE  ALT=\"x%2Ax%2A%28x-3%29%2A16%2A%283x+-+9+%2B+2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Inside the parentheses I  combined the 3x and 2x into 5x so that the expression became:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Ax%2A%28x-3%29%2A16%2A%285x+-9%29 ALIGN=MIDDLE  ALT=\"x%2Ax%2A%28x-3%29%2A16%2A%285x+-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "I then substituted x^2 for x*x and moved the 16 to the front of the line to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%2Ax%5E2%2A%28x-3%29%2A%285x+-9%29 ALIGN=MIDDLE  ALT=\"16%2Ax%5E2%2A%28x-3%29%2A%285x+-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and that's the answer that you were looking for. Hope this helps you to see a way to do this problem.<BR>\n" );
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document.write( "Regarding your question: When we factor (4x-12)^2 it is just (4x-12)(4x-12) instead of (4x-12)(4x+12)? According to the factoring formula \"Difference of squares\", (A-B)^2 = (A-B)(A+B). Guess this is where i got lost but don't understand when you don't foil out these binomials. <BR>\n" );
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document.write( "You're a little confused about factoring the difference of squares. The factoring formula says:<BR>\n" );
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document.write( "A^2 - B^2 = (A - B)*(A + B)<BR>\n" );
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document.write( "Notice the left side of this rule says A^2 - B^2. (A - B)^2 which for this problem is (4x - 12)^2 is not the same as A^2 - B^2. If you have (4x - 12)^2 and you square it by FOIL multiplying (4x - 12)*(4x - 12) you get that:<BR>\n" );
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document.write( "(4x - 12)^2 = 16x^2 - 96x + 144.<BR>\n" );
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document.write( "In general, when you have (A - B)^2 and you square it the result is A^2 - 2AB + B^2 and you can see that it is different from A^2 - B^2 because it has a middle term -2AB and also because it has a positive B^2 instead of a minus B^2. Hope this helps you to see why the difference of squares rule for factoring does not apply to (4x - 12)^2. It would apply if we had (4x)^2 - (12)^2, but that was not the case in this problem. </SPAN>\n" );
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