document.write( "Question 552497: The question is:\r
\n" ); document.write( "\n" ); document.write( "Show that \"%282x%2B1%29\" is a factor of the polynomial \"+2x%5E3%2B5x%5E2%2B8x%2B3\". Hence express \r
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\n" ); document.write( "\n" ); document.write( "I know how to show that 2x+1 is a factor of the polynomial.. I don't know how to express that into partial fractions.\r
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Algebra.Com's Answer #360400 by AnlytcPhil(1806)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"%282x%5E4%2B5x%5E3%2B11x%5E2-15x-3%29%2F%282x%5E3%2B5x%5E2%2B8x%2B3%29\"\r\n" );
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document.write( "Since the numerator has a higher degree than the denominator, before we can do\r\n" );
document.write( "partial fractions, we must first do the long division to get a rational\r\n" );
document.write( "expression which has a numerator of lower degree than the denominator:\r\n" );
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document.write( "                                        x + 0                                \r\n" );
document.write( "2x³ + 5x² + 8x + 3)2x4 + 5x³ + 11x² - 15x - 3\r\n" );
document.write( "                   2x4 + 5x³ +  8x² +  3x\r\n" );
document.write( "                         0x³ +  3x² - 18x - 3\r\n" );
document.write( "                         0x³ +  0x² +  0x + 0\r\n" );
document.write( "                                3x² - 18x - 3\r\n" );
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document.write( "\"%282x%5E4%2B5x%5E3%2B11x%5E2-15x-3%29%2F%282x%5E3%2B5x%5E2%2B8x%2B3%29\" = x + \"%283x%5E2-18x-3%29%2F%282x%5E3%2B5x%5E2%2B8x%2B3%29\"\r\n" );
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document.write( "We'll break \"%283x%5E2-18x-3%29%2F%282x%5E3%2B5x%5E2%2B8x%2B3%29\" into partial fractions.  Then\r\n" );
document.write( "we'll come back and add the x quotient to it.\r\n" );
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document.write( "Now we factor the denominator.  We are told that 2x+1 is a factor, so we\r\n" );
document.write( "divide that into the denominator\r\n" );
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document.write( "              x² + 2x + 3      \r\n" );
document.write( "2x + 1)2x³ + 5x² + 8x + 3\r\n" );
document.write( "       2x³ +  x²\r\n" );
document.write( "             4x² + 8x\r\n" );
document.write( "             4x² + 2x\r\n" );
document.write( "                   6x + 3\r\n" );
document.write( "                   6x + 3 \r\n" );
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document.write( "x²+2x+3 does not factor, so we have \r\n" );
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document.write( "\"%283x%5E2-18x-3%29%2F%282x%5E3%2B5x%5E2%2B8x%2B3%29\" = \"%283x%5E2-18x-3%29%2F%28%282x%2B1%29%28x%5E2%2B2x%2B3%29%29\"\r\n" );
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document.write( "to break into partial fractions:\r\n" );
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document.write( "\"%283x%5E2-18x-3%29%2F%28%282x%2B1%29%28x%5E2%2B2x%2B3%29%29\" = \"A%2F%282x%2B1%29\" + \"%28Bx%2BC%29%2F%28x%5E2%2B2x%2B3%29\"\r\n" );
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document.write( "Multiply through by the LCD\r\n" );
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document.write( "3x² - 18x - 3 = A(x² + 2x + 3) + (Bx + C)(2x + 1)\r\n" );
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document.write( "This must be true for all values of x, so we substitute various values for x:\r\n" );
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document.write( "Substituting x=0\r\n" );
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document.write( "3(0)² - 18(0) - 3 = A((0)² + 2(0) + 3) + (B(0) + C)(2(0) + 1)\r\n" );
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document.write( "               -3 = 3A + C \r\n" );
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document.write( "Substituting x=1\r\n" );
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document.write( "3(1)² - 18(1) - 3 = A((1)² + 2(1) + 3) + (B(1) + C)(2(1) + 1)\r\n" );
document.write( "       3 - 18 - 3 = A(1 + 2 + 3) + (B + C)(2 + 1)\r\n" );
document.write( "              -18 = A(6) + (B + C)(3)\r\n" );
document.write( "              -18 = 6A + 3B + 3C\r\n" );
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document.write( "Substituting x=-1\r\n" );
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document.write( "3(-1)² - 18(-1) - 3 = A((-1)² + 2(-1) + 3) + (B(-1) + C)(2(-1) + 1)\r\n" );
document.write( "       3 + 18 - 3 = A(1 - 2 + 3) + (-B + C)(-2 + 1)\r\n" );
document.write( "               18 = A(2) + (-B + C)(-1)\r\n" );
document.write( "               18 = 2A + B - C\r\n" );
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document.write( "So we solve the system of three equations:\r\n" );
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document.write( "-3 = 3A + C\r\n" );
document.write( "-18 = 6A + 3B + 3C\r\n" );
document.write( "18 = 2A + B - C\r\n" );
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document.write( "and get A=3, B=0, C=-12\r\n" );
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document.write( "So\r\n" );
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document.write( "\"%283x%5E2-18x-3%29%2F%28%282x%2B1%29%28x%5E2%2B2x%2B3%29%29\" = \"3%2F%282x%2B1%29\" + \"%280x-12%29%2F%28x%5E2%2B2x%2B3%29\"\r\n" );
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document.write( " \"%283x%5E2-18x-3%29%2F%28%282x%2B1%29%28x%5E2%2B2x%2B3%29%29\" = \"3%2F%282x%2B1%29\" - \"12%2F%28x%5E2%2B2x%2B3%29\"\r\n" );
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document.write( "Therefore going back to the original problem:\r\n" );
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document.write( "\"%282x%5E4%2B5x%5E3%2B11x%5E2-15x-3%29%2F%282x%5E3%2B5x%5E2%2B8x%2B3%29\" = x + \"3%2F%282x%2B1%29\" - \"12%2F%28x%5E2%2B2x%2B3%29\"\r\n" );
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document.write( "Edwin
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