document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190174>Question 1190174</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In factoring using the difference of cubes formula, if you have the problem 1+8x^3 do you need to switch it to 8x^3+1 to work or can you solve it the way it is written? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #821743 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=821743\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Either way works perfectly fine. This is because adding any two numbers can be done in any order you want.<BR>\n" );
document.write( "A+B = B+A<BR>\n" );
document.write( "Furthermore, 8x^3+1 will factor the same way as 1+8x^3\r<BR>\n" );
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document.write( "I should point out however, that the 8x^3+1 is a sum of cubes (not a difference of cubes). \r<BR>\n" );
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document.write( "The sum of cubes factoring formula is<BR>\n" );
document.write( "a^3 + b^3 = (a+b)(a^2 - ab + b^2)\r<BR>\n" );
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document.write( "One example of using that formula<BR>\n" );
document.write( "a^3 + b^3 = (a+b)(a^2 - ab + b^2)<BR>\n" );
document.write( "(4x)^3 + (2)^3 = (4x+2)((4x)^2 - (4x)*2 + 2^2)<BR>\n" );
document.write( "64x^3 + 8 = (4x+2)(16x^2 - 8x + 4)\r<BR>\n" );
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document.write( "Notice how swapping the roles of 'a' and b doesn't matter<BR>\n" );
document.write( "a^3 + b^3 = (a+b)(a^2 - ab + b^2)<BR>\n" );
document.write( "2^3 + (4x)^3 = (2+4x)(2^2 - 2*4x + (4x)^2)<BR>\n" );
document.write( "8 + 64x^3 = (2+4x)(4 - 8x + 16x^2)<BR>\n" );
document.write( "8 + 64x^3 = (4x+2)(16x^2 - 8x + 4)<BR>\n" );
document.write( "Showing we get the same exact factorization as earlier.\r<BR>\n" );
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document.write( "A related factoring formula is the difference of cubes factoring formula<BR>\n" );
document.write( "a^3 - b^3 = (a-b)(a^2 + ab + b^2)<BR>\n" );
document.write( "In this case, order does matter because a^3-b^3 is not the same as b^3-a^3 (much like how 10-3 = 7 is not the same as 3-10 = -7)\r<BR>\n" );
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document.write( "An example:<BR>\n" );
document.write( "a^3 - b^3 = (a-b)(a^2 + ab + b^2)<BR>\n" );
document.write( "(7x)^3 - 5^3 = (7x-5)((7x)^2 + 7x*5 + 5^2)<BR>\n" );
document.write( "343x^3 - 125 = (7x-5)(49x^2 + 35x + 25)\r<BR>\n" );
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document.write( "If we swapped 'a' and b, then,<BR>\n" );
document.write( "a^3 - b^3 = (a-b)(a^2 + ab + b^2)<BR>\n" );
document.write( "(5)^3 - (7x)^3 = (5-7x)((5)^2 + 5*7x + (7x)^2)<BR>\n" );
document.write( "125 - 343x^3 = (5-7x)(25 + 35x + 49x^2)<BR>\n" );
document.write( "125 - 343x^3 = (5-7x)(49x^2 + 35x + 25)<BR>\n" );
document.write( "It's very close to the earlier factorization in the previous paragraph. <BR>\n" );
document.write( "However, the key difference is that the 7x-5 is not the same as 5-7x.<BR>\n" );
document.write( "Therefore, the two factorizations aren't the same.<BR>\n" );
document.write( "Again it's all because the order of subtraction matters.<BR>\n" );
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