document.write( "Question 1207322: f (x) is polynomial function, f '(x) + int f (x) dx = x ^4 + 13 x ^2 + 2. Find f (x) \n" ); document.write( "
Algebra.Com's Answer #845131 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"int%28%22f%28x%29%22%2Cdx%29\"\"%22%22=%22%22\"\r\n" );
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document.write( "Since the sum of those must be identically equal to \"x%5E4%2B13x%5E2%2B2\",\r\n" );
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document.write( "then the largest power of x that occurs in the sum, which is n+1, must be\r\n" );
document.write( "the largest power that occurs in \"x%5E4%2B13x%5E2%2B2\", which is 4. Therefore,\r\n" );
document.write( "n+1 = 4 and n=3.  Substituting (and eliminating the 3 dots since they\r\n" );
document.write( "are no longer needed):\r\n" );
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document.write( "\"%22f%28x%29%22=a%5B0%5Dx%5E3%2Ba%5B1%5Dx%5E%283-1%29%2Ba%5B3-1%5Dx%2Ba%5B3%5D\"\r\n" );
document.write( "\"%22f%27%28x%29%22=3%2Aa%5B0%5Dx%5E%283-1%29%2B%283-1%29%2Aa%5B1%5Dx%5E%283-2%29%2Ba%5B3-1%5D\"\r\n" );
document.write( "\"int%28%22f%28x%29%22%2Cdx%29\"\"%22%22=%22%22\"\r\n" );
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document.write( "Simplifying,\r\n" );
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document.write( "\"%22f%28x%29%22=a%5B0%5Dx%5E3%2Ba%5B1%5Dx%5E2%2Ba%5B2%5Dx%2Ba%5B3%5D\"\r\n" );
document.write( "\"%22f%27%28x%29%22=3%2Aa%5B0%5Dx%5E2%2B2%2Aa%5B1%5Dx%2Ba%5B2%5D\"\r\n" );
document.write( "\"int%28%22f%28x%29%22%2Cdx%29\"\"%22%22=%22%22\"\r\n" );
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document.write( "Since the coefficients of x3 and x are 0 in \"x%5E4%2B13x%5E2%2B2\",\r\n" );
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document.write( "\"a%5B0%5Dx%5E3%2Bexpr%281%2F3%29%2Aa%5B1%5Dx%5E3=0\" and \"a%5B2%5Dx%2B2a%5B1%5Dx%2Ba%5B3%5D=0\".\r\n" );
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document.write( "Since the coefficient of x2 is 13, \"a%5B1%5Dx%5E2%2B3a%5B0%5Dx%5E2%2Bexpr%281%2F2%29a%5B2%5Dx%5E2=13\" \r\n" );
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document.write( "and since its constant term is 2, \"a%5B3%5D%2Ba%5B2%5D%2BC=2\"\r\n" );
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document.write( "So we have the system of equations\r\n" );
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document.write( "Your teacher must have meant for you to take the constant \r\n" );
document.write( "of integration to be 0.  Otherwise you have 5 unknowns but \r\n" );
document.write( "only 4 equations.  I'll use my system solver to solve that.  \r\n" );
document.write( "I get: \r\n" );
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document.write( "\"system%28a%280%29=-1%2F3%2Ca%281%29=1%2Ca%282%29=26%2Ca%283%29=-26%29\"\r\n" );
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document.write( "So that makes\r\n" );
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document.write( "\"%22f%28x%29%22=expr%28-1%2F3%29x%5E3%2Bx%5E2%2B26x-26\"\r\n" );
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document.write( "Edwin
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