document.write( "Question 1042243: The book is not making any sense to me, please help!\r
\n" ); document.write( "\n" ); document.write( "Perform the long division and write the partial fraction decomposition of the remainder term.\r
\n" ); document.write( "\n" ); document.write( "(x^4 - x^2 + 2)/(x^3 - x^2)
\n" ); document.write( "

Algebra.Com's Answer #657234 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "\"%28x%5E4+-+x%5E2+%2B+2%29%2F%28x%5E3+-+x%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "Put in zero placeholders for missing powers of x\r\n" );
document.write( "\r\n" );
document.write( "\"%28x%5E4+%2B0x%5E3-+x%5E2+%2B0x%2B+2%29%2F%28x%5E3+-+x%5E2%2B0x%2B0%29\"\r\n" );
document.write( "\r\n" );
document.write( "                       x+1\r\n" );
document.write( "x³-x²+0x+0)x⁴+0x³- x²+0x+2\r\n" );
document.write( "           x⁴- x³+0x²+0x\r\n" );
document.write( "               x³- x²+0x+2\r\n" );
document.write( "               x³- x²+0x+0\r\n" );
document.write( "                  0x²+0x+2\r\n" );
document.write( "\r\n" );
document.write( "So \r\n" );
document.write( "\r\n" );
document.write( "\"%28x%5E4+-+x%5E2+%2B+2%29%2F%28x%5E3+-+x%5E2%29\"\"%22%22=%22%22\"\"x%2B1%2B2%2F%28x%5E3-x%5E2%29\"\"%22%22=%22%22\"\r\n" );
document.write( "\r\n" );
document.write( "\"x%2B1+%2B+2%5E%22%22%2F%28x%5E2%28x-1%29%29\"\"%22%22=%22%22\"\"x%2B1%2BA%5E%22%22%2Fx%5E%22%22%2BB%5E%22%22%2Fx%5E2%2BC%5E%22%22%2F%28x-1%29%5E%22%22\"\r\n" );
document.write( "\r\n" );
document.write( "Just taking the fraction parts:\r\n" );
document.write( "\r\n" );
document.write( "\"2%2F%28x%5E2%28x-1%29%29\"\"%22%22=%22%22\"\"A%5E%22%22%2Fx%5E%22%22%2BB%5E%22%22%2Fx%5E2%2BC%5E%22%22%2F%28x-1%29%5E%22%22\"\r\n" );
document.write( "\r\n" );
document.write( "Multiply through by the LCD = x²(x-1)\r\n" );
document.write( "\r\n" );
document.write( "\"2\"\"%22%22=%22%22\"\"Ax%28x-1%29%2BB%28x-1%29%2BCx%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "If we let x=0, 2 terms on the right will become 0:\r\n" );
document.write( "\r\n" );
document.write( "\"2\"\"%22%22=%22%22\"\"A%2A0%280-1%29%2BB%280-1%29%2BC%2A0%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "\"2\"\"%22%22=%22%22\"\"-B\"\r\n" );
document.write( "\r\n" );
document.write( "\"B\"\"%22%22=%22%22\"\"-2\", substituting:\r\n" );
document.write( "\r\n" );
document.write( "\"2\"\"%22%22=%22%22\"\"Ax%28x-1%29-2%28x-1%29%2BCx%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "If we let x=1, 2 terms on the right will become 0\r\n" );
document.write( "\r\n" );
document.write( "\"2\"\"%22%22=%22%22\"\"A%2A1%281-1%29-2%281-1%29%2BC%2A1%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "\"2\"\"%22%22=%22%22\"\"A%280%29-2%280%29%2BC%2A1\"\r\n" );
document.write( "\r\n" );
document.write( "\"2\"\"%22%22=%22%22\"\"C\", substituting:\r\n" );
document.write( "\r\n" );
document.write( "\"2\"\"%22%22=%22%22\"\"Ax%28x-1%29-2%28x-1%29%2B2x%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "Now we can let x be any number, say choose easy x=2\r\n" );
document.write( "\r\n" );
document.write( "\"2\"\"%22%22=%22%22\"\"A%2A2%282-1%29-2%282-1%29%2B2%2A2%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "\"2\"\"%22%22=%22%22\"\"A%2A2%281%29-2%281%29%2B2%2A4\"\r\n" );
document.write( "\r\n" );
document.write( "\"2\"\"%22%22=%22%22\"\"2A-2%2B8\"\r\n" );
document.write( "\r\n" );
document.write( "\"2\"\"%22%22=%22%22\"\"2A%2B6\"\r\n" );
document.write( "\r\n" );
document.write( "\"-4\"\"%22%22=%22%22\"\"2A\"\r\n" );
document.write( "\r\n" );
document.write( "\"-2\"\"%22%22=%22%22\"\"A\", \r\n" );
document.write( "\r\n" );
document.write( "substitute A=-2, B=-2, and C=2 in\r\n" );
document.write( "\r\n" );
document.write( "\"%28x%5E4+-+x%5E2+%2B+2%29%2F%28x%5E3+-+x%5E2%29\"\"%22%22=%22%22\"\"x%2B1+%2B+%28x%5E2%2B2%29%2F%28x%5E2%28x-1%29%29\"\"%22%22=%22%22\"\"x%2B1%2BA%5E%22%22%2Fx%5E%22%22%2BB%5E%22%22%2Fx%5E2%2BC%5E%22%22%2F%28x-1%29%5E%22%22\"\"%22%22=%22%22\"\r\n" );
document.write( "\r\n" );
document.write( "\"x%2B1-2%5E%22%22%2Fx%5E%22%22-2%5E%22%22%2Fx%5E2%2B2%5E%22%22%2F%28x-1%29%5E%22%22\"\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );