document.write( "Question 1165974: Let f(x) be a quadratic polynomial such that f(-4) = -22, f(-1)=2, and f(2)=-1. Let g(x) = f(x)^{16}. Find the sum of the coefficients of the terms in g(x) with even exponents. (For example, the sum of the coefficients of the terms in -7x^3 + 4x^2 + 10x - 5 with even exponents is (4) + (-5) = -1.)
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Algebra.Com's Answer #790447 by ikleyn(52814) $\"\"$ $\"About$

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\n" ); document.write( "Let f(x) be a quadratic polynomial such that f(-4) = -22, f(-1)=2, and f(2)=-1. Let g(x) = f(x)^{16}.
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\n" ); document.write( "\n" ); document.write( "(For example, the sum of the coefficients of the terms in -7x^3 + 4x^2 + 10x - 5 with even exponents is (4) + (-5) = -1.)
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\n" ); document.write( "\n" ); document.write( "The steps are as follows:\r
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document.write( "    1)  First we find the quadratic polynomial  f(x) via its coefficients  f(x) = ax^2 + bx + c.\r\n" );
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document.write( "    2)  Then we find the value of f(1),   which is the sum of its coefficients f(1) = a + b + c.\r\n" );
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document.write( "    3)  Next, we find the value of f(-1), which is an ALTERNATE sum of its coefficients  f(-1) = a - b + c.\r\n" );
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document.write( "    4)  Finally, we will find [f(1)]^16 = (a + b + c)^16  and  [f(-1)]^16 = (a - b + c)^16.\r\n" );
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document.write( "        Then we note that the sum of the coefficients of the terms in g(x) with even exponents is exactly half of the sum  [f(1)]^16 + [f(-1)]^16,\r\n" );
document.write( "        since the odd exponent coefficients will cancel each other when summing.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Now I will implement these steps.\r
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document.write( "Let f(x) = ax^2 + bx + c.\r\n" );
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document.write( "First, we want to find coefficients \"a\", \"b\" and \"c\".\r\n" );
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document.write( "We have  these equations:\r\n" );
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document.write( "    at x= -4:  16a - 4b + c = -22\r\n" );
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document.write( "    at x= -1:    a -  b + c =   2\r\n" );
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document.write( "    at x=  2:   4a + 2b + c =  -1\r\n" );
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document.write( "You can solve the system by any method you want. You will get then   a = -3/2,  b= 1/2,  c= 4.\r\n" );
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document.write( "Hence, the value of f(1) is   = 3.\r\n" );
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document.write( "       The value   f(-1) is just given by the condition  f(-1) = 2.\r\n" );
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document.write( "It implies that the sum  [f(1)]^16 + [f(-1)]^16  is   + .\r\n" );
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document.write( "Hence, the answer to the problem's question is   = ..\r\n" );
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document.write( "ANSWER.  The sum of the coefficients of the terms in g(x) with even exponents is   = .\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "Usually, such problems are considered as \"coffin\" problems, meaning that the average school student does not know how to solve them.\r
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\n" ); document.write( "\n" ); document.write( "Only those are able to solve, who know this trick with alternate sums of the coefficients of polynomials.\r
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\n" ); document.write( "\n" ); document.write( "Those who study Math from Math circles, know it . . . \r
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