P₁(x) = x² + (k-29)x - k and, P₂(x) = 2x²+(2k-43)x + k, where each of P₁(x) and P₂(x) is a factor of a cubic polynomial P(x).

Determine the maximum value of k.

first, assume that P1(x) and P2(x) have no roots in common.  Since any root of either P1(x) or P2(x) is a root of P(x) then that would mean P(x) would have 4 distinct roots.  However it is a cubic polynomial and thus can have at most 3 distinct roots.  Thus P1(x) and P2(x) have at least one root in common.

the roots of P1(x) are:
a1=1/2 (29 - k - Sqrt[841 - 54 k + k^2])
a2=1/2 (29 - k + Sqrt[841 - 54 k + k^2])
the roots of P2(x) are:
b1=1/2 (43 - 2 k - Sqrt[1849 - 176 k + 4 k^2])
b2=1/2 (43 - 2 k + Sqrt[1849 - 176 k + 4 k^2])

there are 4 possibilities for the common root:
a1=b1, a1=b2
a2=b1, a2=b2
solving each of these for k gives:

a1=b1 gives k=0 which gives the 3 zeros of the polynomial as 0 29 and 43 making
P(x)=x(x-29)(x-43)=x^3-72x^2+1247x

a1=b2 gives k=(1/3)*(59+sqrt(457)) which gives the 3 roots
{1/2 (29 + 1/3 (-59 - Sqrt[457]) - Sqrt[
    841 - 18 (59 + Sqrt[457]) + 1/9 (59 + Sqrt[457])^2]),
 1/2 (29 + 1/3 (-59 - Sqrt[457]) + Sqrt[
    841 - 18 (59 + Sqrt[457]) + 1/9 (59 + Sqrt[457])^2]),
 1/2 (43 - 2/3 (59 + Sqrt[457]) - Sqrt[
    1849 - 176/3 (59 + Sqrt[457]) + 4/9 (59 + Sqrt[457])^2])}
and gives
P(x)=1/18 (-265 - 107 Sqrt[457] - 59 Sqrt[1241 - 56 Sqrt[457]] - Sqrt[
   457 (1241 - 56 Sqrt[457])] + (868 - 73 Sqrt[457] -
      28 Sqrt[1241 - 56 Sqrt[457]] + Sqrt[
      457 (1241 - 56 Sqrt[457])]) x +
   3 (-67 + 4 Sqrt[457] + Sqrt[1241 - 56 Sqrt[457]]) x^2 + 18 x^3)

a2=b1 gives no solution for k

a2=b2 gives k=(1/3)*(59-Sqrt[457]) which gives the 3 zeros
{1/2 (29 + 1/3 (-59 + Sqrt[457]) - Sqrt[
    841 - 18 (59 - Sqrt[457]) + 1/9 (59 - Sqrt[457])^2]),
 1/2 (29 + 1/3 (-59 + Sqrt[457]) + Sqrt[
    841 - 18 (59 - Sqrt[457]) + 1/9 (59 - Sqrt[457])^2]),
 1/2 (43 - 2/3 (59 - Sqrt[457]) - Sqrt[
    1849 - 176/3 (59 - Sqrt[457]) + 4/9 (59 - Sqrt[457])^2])}
and
P(x)=1/18 (-265 + 107 Sqrt[457] - 59 Sqrt[1241 + 56 Sqrt[457]] + Sqrt[
   457 (1241 + 56 Sqrt[457])] - (-868 - 73 Sqrt[457] +
      28 Sqrt[1241 + 56 Sqrt[457]] + Sqrt[
      457 (1241 + 56 Sqrt[457])]) x -
   3 (67 + 4 Sqrt[457] - Sqrt[1241 + 56 Sqrt[457]]) x^2 + 18 x^3)

since (1/3)*(59+sqrt(457))>(1/3)*(59-sqrt(457))>0
the maximum value for k is (1/3)*(59+sqrt(457))

However, the only integer value for k is 0.

Posted by Daniel on 2013-09-08 21:42:44