document.write( "Question 1183627: The expression x^3 + ax^2 + bx + 3 is exactly divisible by x+3 but it leaves a remainder of 91 when divided by x-4. What is the remainder when it is divided by x+2? \n" ); document.write( "

Algebra.Com's Answer #814034 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "The expression x^3 + ax^2 + bx + 3 is exactly divisible by x+3
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\n" ); document.write( "\n" ); document.write( " The solution is to apply the Remainder theorem several times.\r
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document.write( "(1)  Due to the Remainder theorem, first condition means that the numner -3 is the root of the given polynomial.\r\n" );
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document.write( "     It gives you this equation\r\n" );
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document.write( "         (-3)^3 + a*(-3)^2 + b*(-3) + 3 = 0,\r\n" );
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document.write( "     which is equivalent to\r\n" );
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document.write( "         9a - 3b = 24,   or    3a - b = 8.      (1)\r\n" );
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document.write( "(2)  Due to the same theorem, second condition means that the value of the given polynomial is 91 at x= 4.\r\n" );
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document.write( "     It gives you this equation\r\n" );
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document.write( "         4^3 + a*4^2 + b*4 + 3 = 91,\r\n" );
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document.write( "     which is equivalent to\r\n" );
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document.write( "         16a + 4b = 24,   or   4a + b = 6.     (2)\r\n" );
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document.write( "(3)  Adding equations(1) and (2), we get  7a = 14,  a = 14/7 = 2.\r\n" );
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document.write( "     Then from (1),  3*2 - b = 8,  b = 6 - 8 = -2.\r\n" );
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document.write( "     So, the polynomial is  p(x) = x^3 + 2*x^2 - 2*x + 3.\r\n" );
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document.write( "(4)  Due to the Remainder theorem, the remainder of the polynomial when it is divided by x+2 is the value of the polynomial at x= -2, i.e.\r\n" );
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document.write( "         (-2)^3 + 2*(-2)^2 -2*(-2) + 3 = 7.      ANSWER\r\n" );
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