document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=76395>Question 76395</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Have tried many ways to factor this: 25x^3y-78x^2y^2+9xy^3=   I hope one of you helps me with this.  Thanks. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #54807 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=54807\">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=25x%5E3y-78x%5E2y%5E2%2B9xy%5E3 ALIGN=MIDDLE  ALT=\"25x%5E3y-78x%5E2y%5E2%2B9xy%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note that every term contains an x, so you can factor it out to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%2825x%5E2y+-+78xy%5E2+%2B+9y%5E3%29 ALIGN=MIDDLE  ALT=\"x%2A%2825x%5E2y+-+78xy%5E2+%2B+9y%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice also that ever term in the parentheses contains a y so y can also be factored<BR>\n" );
document.write( "out. When you do that factoring the expression becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Ay%2A%2825x%5E2+-+78xy+%2B+9y%5E2%29 ALIGN=MIDDLE  ALT=\"x%2Ay%2A%2825x%5E2+-+78xy+%2B+9y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Factoring the polynomial in the parentheses is a little tricky depending on where you are<BR>\n" );
document.write( "in your math lessons. If you have not yet studied the quadratic formula, you can use trial<BR>\n" );
document.write( "and error along with the following identity:<BR>\n" );
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document.write( "(ax + by)(cx + dy) = (ac)x^2 + (ad+bc)xy + (bd)y^2<BR>\n" );
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document.write( "By comparing the right side of this identity with the polynomial in the parentheses, <BR>\n" );
document.write( "you can see that a*c must be 25. This presents several possibilities:<BR>\n" );
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document.write( " a = 25 and c = 1;  or a = 1 and c = 25;  or a = 5 and c = 5.<BR>\n" );
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document.write( "And also by comparing, you can also see that b*d must equal 9.  So possible combinations <BR>\n" );
document.write( "are:<BR>\n" );
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document.write( " b = 9 and c = 1; or b = 1 and c = 9; or b = 3 and c = 3<BR>\n" );
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document.write( "and you must take into consideration whether the signs on these terms are + or -.<BR>\n" );
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document.write( "Now you must experiment with the possibilities to find the combination of a, b, c, and d<BR>\n" );
document.write( "that will result in (a*d + b*c) equaling -78.<BR>\n" );
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document.write( "I played around and found that a = 1, c = 25, b = -3, and d = -3 will work because<BR>\n" );
document.write( "a*d = -3 and b*c = -75 and when combined these two produce -78.<BR>\n" );
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document.write( "Now you can substitute these values for a,b,c, and d into the factored form of (ax + by)(cx + dy). <BR>\n" );
document.write( "When you do you get the two factors to be (x - 3y)(25x - 3y).  Substituting these factors <BR>\n" );
document.write( "into where we left off we get the complete factored form to be:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy%28x-3y%29%2825x-3y%29 ALIGN=MIDDLE  ALT=\"xy%28x-3y%29%2825x-3y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can multiply this out, and if you do, you will find that it checks because the product<BR>\n" );
document.write( "is the polynomial that you were originally given to factor.<BR>\n" );
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document.write( "If you are familiar with the quadratic formula you could also apply it to (25x^2 - 78x + 9 = 0)<BR>\n" );
document.write( "and find that the solutions are: x = 3 and x = 3/25.<BR>\n" );
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document.write( "This indicates that (x - 3) and (25x - 3) are factors. Don't forget that the y also comes<BR>\n" );
document.write( "into play.  That means (x - 3y) and (25x - 3y) are the likely factors. This was the clue<BR>\n" );
document.write( "I used to guess which combinations of values for a, b, c, and d, were likely to work.<BR>\n" );
document.write( "This method certainly reduced the amount of work ...<BR>\n" );
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document.write( "Hope this helps you to understand what might be involved in factoring the given polynomial.<BR>\n" );
document.write( "And now that you have the answer, you may be able to find another way of approaching<BR>\n" );
document.write( "this problem. </SPAN>\n" );
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