document.write( "Question 1196483: Given that $\"f%28x%29=5x%5E2-3x%2B7\"$ and $\"f%28g%28x%29%29=%285x%5E4%2F9%29%2B%2817x%5E2%2F3%29%2B21\"$ , find all possible values for the sum of the coefficients in the quadratic function g(x). \n" ); document.write( "

Algebra.Com's Answer #829326 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Given that $\"f%28x%29=5x%5E2-3x%2B7\"$ and $\"f%28g%28x%29%29=%285x%5E4%2F9%29%2B%2817x%5E2%2F3%29%2B21\"$ , find all possible values
\n" ); document.write( "for the sum of the coefficients in the quadratic function g(x).
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\n" ); document.write( "\n" ); document.write( "            The key idea to solve this problem is to use the fact that the sum of coefficients \r
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\n" ); document.write( "\n" ); document.write( "            of any polynomial   p(x) = $\"a%5Bn%5D%2Ax%5En+%2B+a%5Bn-1%5D%2Ax%5E%28n-1%29+%2B+ellipsis+%2B+a%5B1%5D%2Ax+%2B+a%5B0%5D\"$   is the value \r
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\n" ); document.write( "\n" ); document.write( "            of this polynomial p(1) at x = 1, which is quite OBVIOUS.\r
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document.write( "Based on this idea, the sum of the coefficients in the quadratic function g(x) is g(1).\r\n" );
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document.write( "We don't know this value g(1), but from the problem we can calculate the composition  f(g(1))\r\n" );
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document.write( "by substituting x= 1 into the given formula for f(g(x).  We have then\r\n" );
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document.write( "    f(g(1)) =  +  +  =  =  =  = 27 2/9.\r\n" );
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document.write( "Now we can state that the value g(1), which we are looking for, is one of two possible roots of the equation\r\n" );
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document.write( "    f(x) = 27 2/9,  \r\n" );
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document.write( "or one of the two possible roots of the equation\r\n" );
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document.write( "    5x^2 - 3x + 7 = 27 2/9.\r\n" );
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document.write( "In standard quadratic form, this equation is\r\n" );
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document.write( "    5x^2 - 3x - 20  = 0,\r\n" );
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document.write( "or, multiplying all the terms by 9, for convenience,\r\n" );
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document.write( "    45x^2 - 27x - 182 = 0.\r\n" );
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document.write( "To find the roots, use the quadratic formula.\r\n" );
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document.write( "The roots are   =   and   = .\r\n" );
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document.write( "So we conclude that the sum of the coefficients in the quadratic function g(x) is EITHER   = 2  OR   = -1.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "It is a nice Math circle level problem.\r
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