document.write( "Question 1187763: Given that f(x)=x^3+ax^2+bx-3 leaves remainders 1 and -9 when divided by x-1 and x+1. Calculate the values of a and b
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Algebra.Com's Answer #818775 by ikleyn(52800) $\"\"$ $\"About$

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\n" ); document.write( "Given that f(x) = x^3 + ax^2 + bx - 3 leaves remainders 1 and -9 when divided by x-1 and x+1.
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document.write( "Since  f(x) = x^3 + ax^2 + bx - 3 leaves remainder 1 when divided by x-1, it means,\r\n" );
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document.write( "according to the remainder theorem, that  f(1) = 1,   or\r\n" );
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document.write( "    1^3 + a*1^2 + b*1 - 3 = 1,\r\n" );
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document.write( "or\r\n" );
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document.write( "    a + b = 1 - 1 + 3,\r\n" );
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document.write( "    a + b = 3.      (1)\r\n" );
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document.write( "Since  f(x) = x^3 + ax^2 + bx - 3 leaves remainder -9 when divided by x+1, it means,\r\n" );
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document.write( "according to the remainder theorem, that  f(-1) = -9,   or\r\n" );
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document.write( "    (-1)^3 + a*(-1)^2 + b*(-1) - 3 = -9,\r\n" );
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document.write( "or\r\n" );
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document.write( "    a - b = 1 + 3 -9,\r\n" );
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document.write( "    a - b = -5.     (2)\r\n" );
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document.write( "Thus we have two equations (1) and (2) to find two unknowns \"a\" and \"b\".\r\n" );
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document.write( "So we add these equations, and we get  \r\n" );
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document.write( "    2a = 3 + (-5) = -2,  hence,  a = (-2)/2 = -1.\r\n" );
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document.write( "Then from equation (1),  b = 3 - a = 3 - (-1) = 4.\r\n" );
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document.write( "ANSWER.  a= -1;  b= 4.\r\n" );
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