document.write( "Question 470160: What must be added to x^(3) - 3x^(2) -12x +19 so that the result is exactly divisible by x^(2) + x -6? \n" ); document.write( "
Algebra.Com's Answer #322546 by robertb(5830)\"\" \"About 
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we have to find polynomials \"ax%5E3+%2B+bx%5E2+%2B+cx+%2B+d\" to add to \"x%5E3+-+3x%5E2+-12x+%2B19\" so that the sum is divisible by \"x%5E2+%2B+x+-+6+=+%28x%2B3%29%28x+-+2%29\".
\n" ); document.write( "==> -3 and 2 are roots of the divisor \"x%5E2+%2B+x+-+6\".\r
\n" ); document.write( "\n" ); document.write( "Now .\r
\n" ); document.write( "\n" ); document.write( "By the factor theorem,
\n" ); document.write( "after substituting -3 into the preceding polynomial and simplifying, we must have
\n" ); document.write( "27a - 5b + 5c - d = 1;
\n" ); document.write( "after substituting 2 into the preceding polynomial and simplifying, we must have
\n" ); document.write( "8a +4b + 2c + d = 9;\r
\n" ); document.write( "\n" ); document.write( "Adding both equations, we get 7a - b + c = 2, or b = 7a + c - 2.
\n" ); document.write( "Substituting this into 8a +4b + 2c + d = 9, we get d = -36a - 6c + 17.\r
\n" ); document.write( "\n" ); document.write( "The values for a and c are free to vary.
\n" ); document.write( "As an example, let a = c = 1. ==> b = 7*1 + 1 - 2 = 6, and
\n" ); document.write( "d = -36*1 - 6*1 + 17 = -25
\n" ); document.write( "Then \"x%5E3+%2B+6x%5E2+%2B+x+-25\", when added to \"x%5E3+-+3x%5E2+-12x+%2B19\", will be divisible by \"x%5E2+%2B+x+-6\".
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