document.write( "Question 1149382: To Whom it May Concern,\r
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\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 #770711 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
Find the quadratic which has a remainder of -6 when divided by x - 1, a
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document.write( "Let the quadratic polynomial be Ax² + Bx + C\r\n" );
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document.write( "The remainder theorem says this:\r\n" );
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document.write( "If you substitute r into a polynomial, you will get the same answer as you will\r\n" );
document.write( "get if you divide the polynomial by x-r and take the remainder.\r\n" );
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document.write( ">>Find the quadratic which has a remainder of -6 when divided by x - 1<<\r\n" );
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document.write( "So if r=1, then the remainder theorem says this:\r\n" );
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document.write( "If you substitute 1 into a polynomial, you will get the same answer as you will\r\n" );
document.write( "get if you divide the polynomial by x-1 and take the remainder.\r\n" );
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document.write( "So let's substitute 1 into the quadratic polynomial and set it equal to the\r\n" );
document.write( "remainder -6\r\n" );
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document.write( "A(1)² + B(1) + C = -6\r\n" );
document.write( "       A + B + C = -6\r\n" );
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document.write( ">>a remainder of -4 when divided by x - 3<<\r\n" );
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document.write( "So if r=3, then the remainder theorem says this:\r\n" );
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document.write( "If you substitute 3 into a polynomial, you will get the same answer as you will\r\n" );
document.write( "get if you divide the polynomial by x-3 and take the remainder.\r\n" );
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document.write( "So let's substitute 3 into the quadratic polynomial and set it equal to the\r\n" );
document.write( "remainder -4\r\n" );
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document.write( "A(3)² + B(3) + C = -4\r\n" );
document.write( "     9A + 3B + C = -4\r\n" );
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document.write( ">>and no remainder when divided by x + 1<<\r\n" );
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document.write( "Note that x + 1 is the same as x - (-1) \r\n" );
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document.write( "So if r=-1, then the remainder theorem says this:\r\n" );
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document.write( "If you substitute -1 into a polynomial, you will get the same answer as you will\r\n" );
document.write( "get if you divide the polynomial by x+1 and take the remainder.\r\n" );
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document.write( "So let's substitute -1 into the quadratic polynomial and set it equal to the\r\n" );
document.write( "remainder 0.\r\n" );
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document.write( "A(-1)² + B(-1) + C = 0\r\n" );
document.write( "         A - B + C = 9\r\n" );
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document.write( "So we have 3 equations in 3 unknowns:\r\n" );
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document.write( " A +  B + C = -6\r\n" );
document.write( "9A + 3B + C = -4\r\n" );
document.write( " A -  B + C =  0\r\n" );
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document.write( "Solve that and get A=1, B=-3, and C=-4\r\n" );
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document.write( "So Ax² + Bx + C becomes\r\n" );
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document.write( "    x² - 3x - 4  \r\n" );
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document.write( "Checking:\r\n" );
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document.write( "             x - 2\r\n" );
document.write( "x - 1) x² - 3x - 4\r\n" );
document.write( "       x² -  x\r\n" );
document.write( "           -2x - 4\r\n" );
document.write( "           -2x + 2\r\n" );
document.write( "                -6   <--the remainder is -6 when we divide by x - 1.\r\n" );
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document.write( "             x - 0\r\n" );
document.write( "x - 3) x² - 3x - 4\r\n" );
document.write( "       x² - 3x\r\n" );
document.write( "            0x - 4\r\n" );
document.write( "            0x + 0\r\n" );
document.write( "                -4   <--the remainder is -4 when we divide by x - 3.\r\n" );
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document.write( "             x - 4\r\n" );
document.write( "x + 1) x² - 3x - 4\r\n" );
document.write( "       x² +  x\r\n" );
document.write( "           -4x - 4\r\n" );
document.write( "           -4x - 4\r\n" );
document.write( "                 0   <--the remainder is 0 (no remainder) when we divide by x - 1.\r\n" );
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document.write( "Edwin
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