document.write( "Question 542451: What is the numerical coefficient of x^27 in the complete expansion of
\n" ); document.write( "(x^2 + 5x + 2)^5 . (x^2 - 7x + 3)^9.
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #354582 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "We want the numerical coefficient of the term in x27\r\n" );
document.write( "of the complete expansion of (x²+5x+2)5(x²-7x+3)9.\r\n" );
document.write( "\r\n" );
document.write( "(x²+5x+2)5(x²-7x+3)9 equals this product of 14 trinomials:\r\n" );
document.write( "\r\n" );
document.write( "(x²+5x+2)(x²+5x+2)(x²+5x+2)(x²+5x+2)(x²+5x+2)*(x²-7x+3)(x²-7x+3)(x²-7x+3)(x²-7x+3)(x²-7x+3)(x²-7x+3)(x²-7x+3)(x²-7x+3)(x²-7x+3)\r\n" );
document.write( "\r\n" );
document.write( "The complete expansion can be considered as the sum of all products of 14\r\n" );
document.write( "factors where one factor is taken from each of the 14 trinomials.\r\n" );
document.write( "\r\n" );
document.write( "since 27 = 13·2+1 The only way to get a term in x27 is to take the\r\n" );
document.write( "x² term from 13 of the trinomials and the x-term from the remaining one. \r\n" );
document.write( "\r\n" );
document.write( "There will be 14 terms in x27 to combine, of which \r\n" );
document.write( "\r\n" );
document.write( "5 will be of this form  \r\n" );
document.write( "\r\n" );
document.write( "(5x)*(x²)(x²)(x²)(x²)(x²)(x²)(x²)(x²)(x²)(x²)(x²)(x²)(x²) = 5x27\r\n" );
document.write( "\r\n" );
document.write( "with the (5x) in the 5 positions 1-5\r\n" );
document.write( "\r\n" );
document.write( "and the other 9 will be of this form:\r\n" );
document.write( "\r\n" );
document.write( "(x²)(x²)(x²)(x²)(x²)*(-7x)*(x²)(x²)(x²)(x²)(x²)(x²)(x²)(x²) = -7x27\r\n" );
document.write( "\r\n" );
document.write( "with the (-7x) in the 9 positions 6-14\r\n" );
document.write( "\r\n" );
document.write( "Therefore the term in x27 is\r\n" );
document.write( "\r\n" );
document.write( "5(5x27) + 9(-7x27) = 25x27 - 63x27 = -38x27\r\n" );
document.write( "\r\n" );
document.write( "The coefficient of that term is -38.\r\n" );
document.write( " \r\n" );
document.write( "The answer is -38.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );