The product of two of the four roots of the equation x⁴–18x³+Kx²+200x–1984=0 is –32. Find the value of K.

From the givens, we have

  x^4 - 18x^3 + K x^2 + 200 x - 1984 =

  (x^2 + A x - 32)(x^2 + B x + 62) =

  x^4 + (A + B)x^3 + (AB + 30)x^2 + 2(31A - 16B)x - 1984

Setting the coefficients equal gives

  A + B = -18

  AB + 30 = K

  31A - 16B = 100

Solving these gives

  A = -4

  B = -14

  K = 86

Posted by Bractals on 2006-05-16 18:18:26