document.write( "Question 1149382: To Whom it May Concern,\r
\n" ); document.write( "\n" ); document.write( "Could you help me with this question:
\n" ); document.write( "Find the quadratic which has a remainder of -6 when divided by x - 1, a
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Algebra.Com's Answer #770717 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " I think that I know a SIMPLER way to solve the problem, comparing with what the two other tutors proposed.\r
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document.write( "Since the polynomial does not give a remainder when divided by (x+1), it (the polynomial) can be written in the form\r\n" );
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document.write( "    p(x) = a*(x-t)*(x+1),   (1)\r\n" );
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document.write( "where \"a\" is the leading coefficient and t is another root of the polynomial.\r\n" );
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document.write( "Thus I need to find only two unknown values of \"a\" and \"t.\r\n" );
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document.write( "For it, use two other given conditions.\r\n" );
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document.write( "The fact that  \"the quadratic has a remainder of -6 when divided by x-1\" means that  p(1) = -6 (the Remainder theorem)\r\n" );
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document.write( "    p(1) = a*(1-t)*(1+1) = -6,   or  a*(1-t)*2  = -6,   or  a*(1-t) = -3,   or\r\n" );
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document.write( "    a - at = -3.      (2)\r\n" );
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document.write( "The fact that  \"the quadratic has a remainder of -4 when divided by x-3\" means that  p(3) = -4 (the Remainder theorem, again)\r\n" );
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document.write( "    p(3) = a*(3-t)*(3+1) = -4,   or  a*(3-t)*4  = -4,   or  a*(3-t) = -1,   or\r\n" );
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document.write( "    3a - at = -1.      (3)\r\n" );
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document.write( "Now subtract equation (2) from equation (3).  You will get\r\n" );
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document.write( "     3a - a = -1 - (-3) = -1 + 3 = 2,  i.e.   2a = 2,   or  a = 1.\r\n" );
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document.write( "Next substitute the value of a= 1 into equation (2).  You will get\r\n" );
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document.write( "     1 - t = -3,  1 + 3 = t,   t = 4.\r\n" );
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document.write( "Hence, the polynomial under the question is\r\n" );
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document.write( "     p(x) = 1*(x-4)*(x+1) = x^2 - 3x - 4.    ANSWER\r\n" );
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document.write( "You may check it on your own that all conditions of the problem are satisfied.\r\n" );
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\n" ); document.write( "\n" ); document.write( "In this way, I make you free from the necessity to solve 3x3-system of equations.\r
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\n" ); document.write( "\n" ); document.write( "My congrats (!)\r
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\n" ); document.write( "\n" ); document.write( "Come to the forum soon again to learn something new (!)\r
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