document.write( "Question 978568: http://www.algebra.com/tutors/students/your-answer.mpl?question=978456\r
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Algebra.Com's Answer #599934 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "I already solved it for you here:\r\n" );
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document.write( "http://www.algebra.com/tutors/students/your-answer.mpl?question=978038\r\n" );
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document.write( "Maybe you thought that since I used long division instead of synthetic\r\n" );
document.write( "division that that was not using the remainder theorem.  But synthetic\r\n" );
document.write( "division is only a shorthand form of long division.  You can actually use\r\n" );
document.write( "synthetic division.  You'll get the same thing.  I thought it might\r\n" );
document.write( "confuse you since there are two variables.  OK, I'll do it with synthetic\r\n" );
document.write( "division:\r\n" );
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document.write( "a⁴-7a²b²+k⁴\r\n" );
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document.write( "You have to put the b's in for they are coefficients of the a's.  Also you\r\n" );
document.write( "must insert 0 terms for the missing powers of a\r\n" );
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document.write( "3b | 1  0b  -7b²  0b³  kb⁴\r\n" );
document.write( "   |    3b   9b²  6b³ 18b⁴  \r\n" );
document.write( "     1  3b   2b2  6b3 (kb⁴+18b⁴) \r\n" );
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document.write( "The remainder must = 0 so that a-3b \r\n" );
document.write( "will be a factor of a⁴-7a²b²+kb⁴.\r\n" );
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document.write( "kb⁴+18b⁴ = 0\r\n" );
document.write( "(k+18)b⁴ = 0\r\n" );
document.write( "k+18=0; b⁴=0\r\n" );
document.write( " k=-18;  b=0  \r\n" );
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document.write( "There is a trivial case for b=0 and k is any number, \r\n" );
document.write( "which we ignore.\r\n" );
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document.write( "So k = -18\r\n" );
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document.write( "a⁴-7a²b²-18b⁴\r\n" );
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document.write( "(a²+2b²)(a²-9b²)\r\n" );
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document.write( "(a²+2b²)(a-3b)(a+3b)    <--- complete factorization\r\n" );
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document.write( "Edwin
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