document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=355705>Question 355705</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Regarding Products of Sums and Differences<BR>\n" );
document.write( "Multiply and simplify (3xy^2+4y)(-3xy^2+4y)\r<BR>\n" );
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document.write( "1) Please show me the steps to solving this. I have tried numberous times and do not get the answer the book shows. \r<BR>\n" );
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document.write( "2)Why is commutative law used? How should I know when to use it? Why are they switched? How should I know on my own when to switch them? The book shows the first step goes to (4y+3xy^2) (4y-3xy^2). I do not understand that at all.\r<BR>\n" );
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document.write( "I appreciate any help.<BR>\n" );
document.write( "Jill </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #254020 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=254020\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Multiply and simplify (3xy^2+4y)(-3xy^2+4y) \r<BR>\n" );
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document.write( "The distributive rule is:\r<BR>\n" );
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document.write( "a * (b + c) = (a*b) + (a*c)\r<BR>\n" );
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document.write( "The more generalized distributive rule is:\r<BR>\n" );
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document.write( "(a+b) + (c + d) = (a*c) + (a*d) + (b*c) + (b*d)\r<BR>\n" );
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document.write( "you can theoretically expand this as many times as you want, i.e. you can also get:\r<BR>\n" );
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document.write( "(a + b + c) * (d + e + f) = (a*d) + (a*e) + (a*f) + (b*d) + (b*e) + (b*f) + (c*d) + (c*e) + c*f)\r<BR>\n" );
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document.write( "An easy way to see this is to give each variable a number.\r<BR>\n" );
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document.write( "Let a = 1, b = 2, c = 3, d = 4, e = 5, f = 6)\r<BR>\n" );
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document.write( "You can see that (a + b + c) = 6 and (d + e + f) = 15 and 6 * 15 = 90\r<BR>\n" );
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document.write( "Start with (1 + 2 + 3) * (4 + 5 + 6) and distribute them according to the laws of distribution.\r<BR>\n" );
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document.write( "You will get:\r<BR>\n" );
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document.write( "(1*4) + (1*5) + (1*6) + (2*4) + (2*5) + (2*6) + (3*4) + (3*5) + (3*6)\r<BR>\n" );
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document.write( "simplify this to get:\r<BR>\n" );
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document.write( "4 + 5 + 6 + 8 + 10 + 12 + 12 + 15 + 18\r<BR>\n" );
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document.write( "add them all up together to get 90.\r<BR>\n" );
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document.write( "The distribution laws work.\r<BR>\n" );
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document.write( "The commutative laws are:\r<BR>\n" );
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document.write( "a*b = b*a\r<BR>\n" );
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document.write( "they can be extended to say:\r<BR>\n" );
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document.write( "a*b*c = a*c*b = b*a*c = b*c*a = c*a*b = c*b*a\r<BR>\n" );
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document.write( "Again, put it into number and you can see that they work.\r<BR>\n" );
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document.write( "let a = 1, b = 2, c = 3\r<BR>\n" );
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document.write( "you can see that 1*2*3 = 6 and 1*3*2 = 6 and 2*1*3 = 6 and 2*3*1 = 6, etc.\r<BR>\n" );
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document.write( "The commutative laws work.\r<BR>\n" );
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document.write( "The associative laws state that:\r<BR>\n" );
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document.write( "a*(b*c) = (a*b)*c\r<BR>\n" );
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document.write( "Again, you can see that this works by substituting number for letters.\r<BR>\n" );
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document.write( "Let a = 1, b = 2, c = 3, and you get:\r<BR>\n" );
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document.write( "1*(2*3) = 1*6 = 6<BR>\n" );
document.write( "(1*2)*3) = 2*3 = 6\r<BR>\n" );
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document.write( "etc.\r<BR>\n" );
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document.write( "The associative laws work.\r<BR>\n" );
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document.write( "Now to your problem.\r<BR>\n" );
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document.write( "Your problem is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283xy%5E2%2B4y%29%28-3xy%5E2%2B4y%29 ALIGN=MIDDLE  ALT=\"%283xy%5E2%2B4y%29%28-3xy%5E2%2B4y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This can be re-written as:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2Ax%2Ay%5E2%2B4%2Ay%29%28-3%2Ax%2Ay%5E2%2B4%2Ay%29 ALIGN=MIDDLE  ALT=\"%283%2Ax%2Ay%5E2%2B4%2Ay%29%28-3%2Ax%2Ay%5E2%2B4%2Ay%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Use the distributive law to break these out.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%283%2Ax%2Ay%5E2%29+%2A+%28-3%2Ax%2Ay%5E2%29%29+%2B+%28%283%2Ax%2Ay%5E2%29+%2A+%284%2Ay%29%29+%2B+%28%284%2Ay%29%2A%28-3%2Ax%2Ay%5E2%29%29+%2B+%28%284%2Ay%29+%2A+%284%2Ay%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To simplify within each set of outer set of parentheses, you would use the associative law.\r<BR>\n" );
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document.write( "Example:\r<BR>\n" );
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document.write( "The associative law say that you can re-write this as:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Ax%2Ay%5E2%2A%28-3%29%2Ax%2Ay%5E2 ALIGN=MIDDLE  ALT=\"3%2Ax%2Ay%5E2%2A%28-3%29%2Ax%2Ay%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The commutative law says that you can re-write this as:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A%28-3%29%2Ax%2Ax%2Ay%5E2%2Ay%5E2 ALIGN=MIDDLE  ALT=\"3%2A%28-3%29%2Ax%2Ax%2Ay%5E2%2Ay%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The associative law says that you can regroup them as:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2A%28-3%29%29+%2A+%28x%2Ax%29+%2A+%28y%5E2%2Ay%5E2%29%29 ALIGN=MIDDLE  ALT=\"%283%2A%28-3%29%29+%2A+%28x%2Ax%29+%2A+%28y%5E2%2Ay%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You then simplify by performing the operations indicated starting from the inner set of parentheses and working out.\r<BR>\n" );
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document.write( "The expression becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-9%29+%2A+%28x%5E2%29+%2A+%28y%5E4%29 ALIGN=MIDDLE  ALT=\"%28-9%29+%2A+%28x%5E2%29+%2A+%28y%5E4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which can be re-written as:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9%2Ax%5E2%2Ay%5E4 ALIGN=MIDDLE  ALT=\"-9%2Ax%5E2%2Ay%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The associative law allowed you to do that.\r<BR>\n" );
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document.write( "You do the same with all the other multiplications within each set of parentheses to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%283xy%5E2%29+%2A+%28-3xy%5E2%29%29+%2B+%28%283xy%5E2%29+%2A+%284y%29%29+%2B+%28%284y%29%2A%28-3xy%5E2%29%29+%2B+%28%284y%29+%2A+%284y%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28-9%2Ax%5E2%2Ay%5E4%29+%2B+%2812%2Ax%2Ay%5E3%29+%2B+%28-12%2Ax%2Ay%5E3%29+%2B+%2816%2Ay%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "you now combine like terms to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++%28-9%2Ax%5E2%2Ay%5E4%29+%2B+%2816%2Ay%5E2%29+ ALIGN=MIDDLE  ALT=\"++%28-9%2Ax%5E2%2Ay%5E4%29+%2B+%2816%2Ay%5E2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "those 2 middle terms canceled out.\r<BR>\n" );
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document.write( "This can also be written as:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++-9x%5E2y%5E4+%2B+16y%5E2+ ALIGN=MIDDLE  ALT=\"++-9x%5E2y%5E4+%2B+16y%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To convince yourself that you did it right, you can substitute numbers for letters and see if you get the same answer with the original equation and the final equation.\r<BR>\n" );
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document.write( "let x = 2 and y = 3.\r<BR>\n" );
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document.write( "the final equation becomes -9*4*81 + 16*9 = -2916 + 144 = -2772\r<BR>\n" );
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document.write( "the original equation of (3xy^2+4y)(-3xy^2+4y) becomes:\r<BR>\n" );
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document.write( "(3*2*9 + 4*3) * (-3*2*9 + 4*3) which becomes:\r<BR>\n" );
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document.write( "66 * -42 which becomes -2771\r<BR>\n" );
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document.write( "We got the same answer so we must have done it right.\r<BR>\n" );
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document.write( "The test is not fool proof but most of the time it works.\r<BR>\n" );
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document.write( "If you're in doubt, then pick some other numbers and go through the exercise again.\r<BR>\n" );
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document.write( "If it comes up good 2 or 3 times when you choose random numbers, then odds are the solution is good. </SPAN>\n" );
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